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- 1.1: Number Systems
- In this section we introduce the number systems that we will work with in the remainder of this text.
- 1.2: Solving Equations
- In this section, we review the equation-solving skills that are prerequisite for successful completion of the material in this text. Before we list the tools used in the equation solving process, let’s make sure that we understand what is meant by the phrase “solve for x.”
- 1.3: Logic
- Two of the most subtle words in the English language are the words “and” and “or.” One has only three letters, the other two, but it is absolutely amazing how much confusion these two tiny words can cause. Our intent in this section is to clear the mystery surrounding these words and prepare you for the mathematics that depends upon a thorough understanding of the words “and” and “or.”
- 1.4: Compound Inequalities
- This section discusses a technique that is used to solve compound inequalities, which is a phrase that usually refers to a pair of inequalities connected either by the word “and” or the word “or.”
- 1.5: Chapter 1 Exercises with Solutions
Find the derivatives of the following expressions:
Try these example problems on your own to check your understanding and then watch the video for a walk-through of the answers.
Theorem 1: If n is an odd integer, then n 2 is odd as well.
Proof. An odd positive integer n can be written as n = 2k + 1, for some integer k ≥ 0. Then
n 2 = (2k+1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) + 1.
Since (2k 2 + 2k) is even and “even plus one is odd”, we can conclude that n 2 is odd.
Theorem 2: Let G = ( V, E ) be a graph. Then sum of the degrees of all vertices is an even integer, i.e.,
∑ v∈V deg(v)
Proof. If you do not see the meaning of this statement, then first try it out for few graphs. The reason why the statement holds is very simple: Each edge contributes 2 to the summation (because an edge is incident on exactly two distinct vertices).
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Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872.
The philosophical impetus behind Frege's logicist programme from the Grundlagen der Arithmetik onwards was in part his dissatisfaction with the epistemological and ontological commitments of then-extant accounts of the natural numbers, and his conviction that Kant's use of truths about the natural numbers as examples of synthetic a priori truth was incorrect.
This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox identifying an inconsistency in Frege's system set out in the Grundgesetze der Arithmetik. Note that naive set theory also suffers from this difficulty.
On the other hand, Russell wrote The Principles of Mathematics in 1903 using the paradox and developments of Giuseppe Peano's school of geometry. Since he treated the subject of primitive notions in geometry and set theory, this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their Principia Mathematica. 
Today, the bulk of extant mathematics is believed to be derivable logically from a small number of extralogical axioms, such as the axioms of Zermelo–Fraenkel set theory (or its extension ZFC), from which no inconsistencies have as yet been derived. Thus, elements of the logicist programmes have proved viable, but in the process theories of classes, sets and mappings, and higher-order logics other than with Henkin semantics, have come to be regarded as extralogical in nature, in part under the influence of Quine's later thought.
Kurt Gödel's incompleteness theorems show that no formal system from which the Peano axioms for the natural numbers may be derived — such as Russell's systems in PM — can decide all the well-formed sentences of that system.  This result damaged Hilbert's programme for foundations of mathematics whereby 'infinitary' theories — such as that of PM — were to be proved consistent from finitary theories, with the aim that those uneasy about 'infinitary methods' could be reassured that their use should provably not result in the derivation of a contradiction. Gödel's result suggests that in order to maintain a logicist position, while still retaining as much as possible of classical mathematics, one must accept some axiom of infinity as part of logic. On the face of it, this damages the logicist programme also, albeit only for those already doubtful concerning 'infinitary methods'. Nonetheless, positions deriving from both logicism and from Hilbertian finitism have continued to be propounded since the publication of Gödel's result.
One argument that programmes derived from logicism remain valid might be that the incompleteness theorems are 'proved with logic just like any other theorems'. However, that argument appears not to acknowledge the distinction between theorems of first-order logic and theorems of higher-order logic. The former can be proven using finistic methods, while the latter — in general — cannot. Tarski's undefinability theorem shows that Gödel numbering can be used to prove syntactical constructs, but not semantic assertions. Therefore, the claim that logicism remains a valid programme may commit one to holding that a system of proof based on the existence and properties of the natural numbers is less convincing than one based on some particular formal system. 
Logicism — especially through the influence of Frege on Russell and Wittgenstein  and later Dummett — was a significant contributor to the development of analytic philosophy during the twentieth century.
Ivor Grattan-Guinness states that the French word 'Logistique' was "introduced by Couturat and others at the 1904 International Congress of Philosophy, and was used by Russell and others from then on, in versions appropriate for various languages." (G-G 2000:501).
Apparently the first (and only) usage by Russell appeared in his 1919: "Russell referred several time [sic] to Frege, introducing him as one 'who first succeeded in "logicising" mathematics' (p. 7). Apart from the misrepresentation (which Russell partly rectified by explaining his own view of the role of arithmetic in mathematics), the passage is notable for the word which he put in quotation marks, but their presence suggests nervousness, and he never used the word again, so that 'logicism' did not emerge until the later 1920s" (G-G 2002:434). 
About the same time as Carnap (1929), but apparently independently, Fraenkel (1928) used the word: "Without comment he used the name 'logicism' to characterise the Whitehead/Russell position (in the title of the section on p. 244, explanation on p. 263)" (G-G 2002:269). Carnap used a slightly different word 'Logistik' Behmann complained about its use in Carnap's manuscript so Carnap proposed the word 'Logizismus', but he finally stuck to his word-choice 'Logistik' (G-G 2002:501). Ultimately "the spread was mainly due to Carnap, from 1930 onwards." (G-G 2000:502).
Symbolic logic: The overt intent of Logicism is to derive all of mathematics from symbolic logic (Frege, Dedekind, Peano, Russell.) As contrasted with algebraic logic (Boolean logic) that employs arithmetic concepts, symbolic logic begins with a very reduced set of marks (non-arithmetic symbols), a few "logical" axioms that embody the "laws of thought", and rules of inference that dictate how the marks are to be assembled and manipulated — for instance substitution and modus ponens (ie from  A materially implies B and  A, one may derive B). Logicism also adopts from Frege's groundwork the reduction of natural language statements from "subject|predicate" into either propositional "atoms" or the "argument|function" of "generalization"—the notions "all", "some", "class" (collection, aggregate) and "relation".
In a logicist derivation of the natural numbers and their properties, no "intuition" of number should "sneak in" either as an axiom or by accident. The goal is to derive all of mathematics, starting with the counting numbers and then the real numbers, from some chosen "laws of thought" alone, without any tacit assumptions of "before" and "after" or "less" and "more" or to the point: "successor" and "predecessor". Gödel 1944 summarized Russell's logicistic "constructions", when compared to "constructions" in the foundational systems of Intuitionism and Formalism ("the Hilbert School") as follows: "Both of these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principal aims of Russell's constructivism" (Gödel 1944 in Collected Works 1990:119).
History: Gödel 1944 summarized the historical background from Leibniz's in Characteristica universalis, through Frege and Peano to Russell: "Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pure logic", whereas Peano "was more interested in its applications within mathematics". But "It was only [Russell's] Principia Mathematica that full use was made of the new method for actually deriving large parts of mathematics from a very few logical concepts and axioms. In addition, the young science was enriched by a new instrument, the abstract theory of relations" (p. 120-121).
Kleene 1952 states it this way: "Leibniz (1666) first conceived of logic as a science containing the ideas and principles underlying all other sciences. Dedekind (1888) and Frege (1884, 1893, 1903) were engaged in defining mathematical notions in terms of logical ones, and Peano (1889, 1894–1908) in expressing mathematical theorems in a logical symbolism" (p. 43) in the previous paragraph he includes Russell and Whitehead as exemplars of the "logicistic school", the other two "foundational" schools being the intuitionistic and the "formalistic or axiomatic school" (p. 43).
Frege 1879 describes his intent in the Preface to his 1879 Begriffsschrift: He started with a consideration of arithmetic: did it derive from "logic" or from "facts of experience"?
"I first had to ascertain how far one could proceed in arithmetic by means of inferences alone, with the sole support of those laws of thought that transcend all particulars. My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed I had to bend every effort to keep the chain of inferences free of gaps . . . I found the inadequacy of language to be an obstacle no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present ideography. Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed" (Frege 1879 in van Heijenoort 1967:5).
Dedekind 1887 describes his intent in the 1887 Preface to the First Edition of his The Nature and Meaning of Numbers. He believed that in the "foundations of the simplest science viz., that part of logic which deals with the theory of numbers" had not been properly argued — "nothing capable of proof ought to be accepted without proof":
In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions of intuitions of space and time, that I consider it an immediate result from the laws of thought . . . numbers are free creations of the human mind . . . [and] only through the purely logical process of building up the science of numbers . . . are we prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind" (Dedekind 1887 Dover republication 1963 :31).
Peano 1889 states his intent in his Preface to his 1889 Principles of Arithmetic:
Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source in the ambiguity of language. ¶ That is why it is of the utmost importance to examine attentively the very words we use. My goal has been to undertake this examination" (Peano 1889 in van Heijenoort 1967:85).
Russell 1903 describes his intent in the Preface to his 1903 Principles of Mathematics:
"THE present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles" (Preface 1903:vi). "A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the philosophy of Dynamics. . . . [From two questions — acceleration and absolute motion in a "relational theory of space"] I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and then, with a view to discovering the meaning of the word any, to Symbolic Logic" (Preface 1903:vi-vii).
Dedekind and Frege: The epistemologies of Dedekind and of Frege seem less well-defined than that of Russell, but both seem accepting as a priori the customary "laws of thought" concerning simple propositional statements (usually of belief) these laws would be sufficient in themselves if augmented with theory of classes and relations (e.g. x R y) between individuals x and y linked by the generalization R.
Dedekind's "free formations of the human mind" in contrast to the "strictures" of Kronecker: Dedekind's argument begins with "1. In what follows I understand by thing every object of our thought" we humans use symbols to discuss these "things" of our minds "A thing is completely determined by all that can be affirmed or thought concerning it" (p. 44). In a subsequent paragraph Dedekind discusses what a "system S is: it is an aggregate, a manifold, a totality of associated elements (things) a, b, c" he asserts that "such a system S . . . as an object of our thought is likewise a thing (1) it is completely determined when with respect to every thing it is determined whether it is an element of S or not.*" (p. 45, italics added). The * indicates a footnote where he states that:
"Kronecker not long ago (Crelle's Journal, Vol. 99, pp. 334-336) has endeavored to impose certain limitations upon the free formation of concepts in mathematics which I do not believe to be justified" (p. 45).
Indeed he awaits Kronecker's "publishing his reasons for the necessity or merely the expediency of these limitations" (p. 45).
Leopold Kronecker, famous for his assertion that "God made the integers, all else is the work of man"  had his foes, among them Hilbert. Hilbert called Kronecker a "dogmatist, to the extent that he accepts the integer with its essential properties as a dogma and does not look back"  and equated his extreme constructivist stance with that of Brouwer's intuitionism, accusing both of "subjectivism": "It is part of the task of science to liberate us from arbitrariness, sentiment and habit and to protect us from the subjectivism that already made itself felt in Kronecker's views and, it seems to me, finds its culmination in intuitionism".  Hilbert then states that "mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker . . ." (p. 479).
Russell as realist: Russell's Realism served him as an antidote to British Idealism,  with portions borrowed from European Rationalism and British empiricism.  To begin with, "Russell was a realist about two key issues: universals and material objects" (Russell 1912:xi). For Russell, tables are real things that exist independent of Russell the observer. Rationalism would contribute the notion of a priori knowledge,  while empiricism would contribute the role of experiential knowledge (induction from experience).  Russell would credit Kant with the idea of "a priori" knowledge, but he offers an objection to Kant he deems "fatal": "The facts [of the world] must always conform to logic and arithmetic. To say that logic and arithmetic are contributed by us does not account for this" (1912:87) Russell concludes that the a priori knowledge that we possess is "about things, and not merely about thoughts" (1912:89). And in this Russell's epistemology seems different from that of Dedekind's belief that "numbers are free creations of the human mind" (Dedekind 1887:31) 
But his epistemology about the innate (he prefers the word a priori when applied to logical principles, cf. 1912:74) is intricate. He would strongly, unambiguously express support for the Platonic "universals" (cf. 1912:91-118) and he would conclude that truth and falsity are "out there" minds create beliefs and what makes a belief true is a fact, "and this fact does not (except in exceptional cases) involve the mind of the person who has the belief" (1912:130).
Where did Russell derive these epistemic notions? He tells us in the Preface to his 1903 Principles of Mathematics. Note that he asserts that the belief: "Emily is a rabbit" is non-existent, and yet the truth of this non-existent proposition is independent of any knowing mind if Emily really is a rabbit, the fact of this truth exists whether or not Russell or any other mind is alive or dead, and the relation of Emily to rabbit-hood is "ultimate" :
"On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. . . . The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. . . . Formally, my premisses are simply assumed but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour." (Preface 1903:viii)
Russell's paradox: In 1902 Russell discovered a "vicious circle" (Russell's paradox) in Frege's Grundgesetze der Arithmetik, derived from Frege's Basic Law V and he was determined not to repeat it in his 1903 Principles of Mathematics. In two Appendices added at the last minute he devoted 28 pages to both a detailed analysis of Frege's theory contrasted against his own, and a fix for the paradox. But he was not optimistic about the outcome:
"In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions requisite for the notion of class. And the contradiction discussed in Chapter x. proves that something is amiss, but what this is I have hitherto failed to discover. (Preface to Russell 1903:vi)"
"Fictionalism" and Russell's no-class theory: Gödel in his 1944 would disagree with the young Russell of 1903 ("[my premisses] allow mathematics to be true") but would probably agree with Russell's statement quoted above ("something is amiss") Russell's theory had failed to arrive at a satisfactory foundation of mathematics: the result was "essentially negative i.e. the classes and concepts introduced this way do not have all the properties required for the use of mathematics" (Gödel 1944:132).
How did Russell arrive in this situation? Gödel observes that Russell is a surprising "realist" with a twist: he cites Russell's 1919:169 "Logic is concerned with the real world just as truly as zoology" (Gödel 1944:120). But he observes that "when he started on a concrete problem, the objects to be analyzed (e.g. the classes or propositions) soon for the most part turned into "logical fictions" . . . [meaning] only that we have no direct perception of them." (Gödel 1944:120)
In an observation pertinent to Russell's brand of logicism, Perry remarks that Russell went through three phases of realism: extreme, moderate and constructive (Perry 1997:xxv). In 1903 he was in his extreme phase by 1905 he would be in his moderate phase. In a few years he would "dispense with physical or material objects as basic bits of the furniture of the world. He would attempt to construct them out of sense-data" in his next book Our knowledge of the External World " (Perry 1997:xxvi).
These constructions in what Gödel 1944 would call "nominalistic constructivism . . . which might better be called fictionalism" derived from Russell's "more radical idea, the no-class theory" (p. 125):
"according to which classes or concepts never exist as real objects, and sentences containing these terms are meaningful only as they can be interpreted as . . . a manner of speaking about other things" (p. 125).
See more in the Criticism sections, below.
The logicism of Frege and Dedekind is similar to that of Russell, but with differences in the particulars (see Criticisms, below). Overall, the logicist derivations of the natural numbers are different from derivations from, for example, Zermelo's axioms for set theory ('Z'). Whereas, in derivations from Z, one definition of "number" uses an axiom of that system — the axiom of pairing — that leads to the definition of "ordered pair" — no overt number axiom exists in the various logicist axiom systems allowing the derivation of the natural numbers. Note that the axioms needed to derive the definition of a number may differ between axiom systems for set theory in any case. For instance, in ZF and ZFC, the axiom of pairing, and hence ultimately the notion of an ordered pair is derivable from the Axiom of Infinity and the Axiom of Replacement and is required in the definition of the Von Neumann numerals (but not the Zermelo numerals), whereas in NFU the Frege numerals may be derived in an analogous way to their derivation in the Grundgesetze.
The Principia, like its forerunner the Grundgesetze, begins its construction of the numbers from primitive propositions such as "class", "propositional function", and in particular, relations of "similarity" ("equinumerosity": placing the elements of collections in one-to-one correspondence) and "ordering" (using "the successor of" relation to order the collections of the equinumerous classes)".  The logicistic derivation equates the cardinal numbers constructed this way to the natural numbers, and these numbers end up all of the same "type" — as classes of classes — whereas in some set theoretical constructions — for instance the von Neumman and the Zermelo numerals — each number has its predecessor as a subset. Kleene observes the following. (Kleene's assumptions (1) and (2) state that 0 has property P and n+1 has property P whenever n has property P.)
"The viewpoint here is very different from that of [Kronecker]'s maxim that 'God made the integers' plus Peano's axioms of number and mathematical induction], where we presupposed an intuitive conception of the natural number sequence, and elicited from it the principle that, whenever a particular property P of natural numbers is given such that (1) and (2), then any given natural number must have the property P." (Kleene 1952:44).
The importance to the logicist programme of the construction of the natural numbers derives from Russell's contention that "That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected" (1919:4). One derivation of the real numbers derives from the theory of Dedekind cuts on the rational numbers, rational numbers in turn being derived from the naturals. While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophical difficulties appear in a logicist derivation of the natural numbers, these problems should be sufficient to stop the program until these are resolved (see Criticisms, below).
One attempt to construct the natural numbers is summarized by Bernays 1930–1931.  But rather than use Bernays' précis, which is incomplete in some details, an attempt at a paraphrase of Russell's construction, incorporating some finite illustrations, is set out below:
For Russell, collections (classes) are aggregates of "things" specified by proper names, that come about as the result of propositions (assertions of fact about a thing or things). Russell analysed this general notion. He begins with "terms" in sentences, which he analysed as follows:
Terms: For Russell, "terms" are either "things" or "concepts": "Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words, unit, individual, and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. is in some sense. A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term and to deny that such and such a thing is a term must always be false" (Russell 1903:43)
Things are indicated by proper names concepts are indicated by adjectives or verbs: "Among terms, it is possible to distinguish two kinds, which I shall call respectively things and concepts the former are the terms indicated by proper names, the latter those indicated by all other words . . . Among concepts, again, two kinds at least must be distinguished, namely those indicated by adjectives and those indicated by verbs" (1903:44).
Concept-adjectives are "predicates" concept-verbs are "relations": "The former kind will often be called predicates or class-concepts the latter are always or almost always relations." (1903:44)
The notion of a "variable" subject appearing in a proposition: "I shall speak of the terms of a proposition as those terms, however numerous, which occur in a proposition and may be regarded as subjects about which the proposition is. It is a characteristic of the terms of a proposition that anyone of them may be replaced by any other entity without our ceasing to have a proposition. Thus we shall say that "Socrates is human" is a proposition having only one term of the remaining component of the proposition, one is the verb, the other is a predicate.. . . Predicates, then, are concepts, other than verbs, which occur in propositions having only one term or subject." (1903:45)
Truth and falsehood: Suppose one were to point to an object and say: "This object in front of me named 'Emily' is a woman." This is a proposition, an assertion of the speaker's belief, which is to be tested against the "facts" of the outer world: "Minds do not create truth or falsehood. They create beliefs . . . what makes a belief true is a fact, and this fact does not (except in exceptional cases) in any way involve the mind of the person who has the belief" (1912:130). If by investigation of the utterance and correspondence with "fact", Russell discovers that Emily is a rabbit, then his utterance is considered "false" if Emily is a female human (a female "featherless biped" as Russell likes to call humans, following Diogenes Laërtius's anecdote about Plato), then his utterance is considered "true".
Classes (aggregates, complexes): "The class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate" (1903 p. 55). Classes can be specified by extension (listing their members) or by intension, i.e. by a "propositional function" such as "x is a u" or "x is v". But "if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential." (1909 p. 66)
Propositional functions: "The characteristic of a class concept, as distinguished from terms in general, is that "x is a u" is a propositional function when, and only when, u is a class-concept." (1903:56)
Extensional versus intensional definition of a class: "71. Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. . . logically the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal."(1903:69)
The definition of the natural numbers Edit
In the Prinicipia, the natural numbers derive from all propositions that can be asserted about any collection of entities. Russell makes this clear in the second (italicized) sentence below.
"In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration. In the second place, the collections having a given number of terms themselves presumably form an infinite collection: it is to be presumed, for example, that there are an infinite collection of trios in the world, for if this were not the case the total number of things in the world would be finite, which, though possible, seems unlikely. In the third place, we wish to define "number" in such a way that infinite numbers may be possible thus we must be able to speak of the number of terms in an infinite collection, and such a collection must be defined by intension, i.e. by a property common to all its members and peculiar to them." (1919:13)
To illustrate, consider the following finite example: Suppose there are 12 families on a street. Some have children, some do not. To discuss the names of the children in these households requires 12 propositions asserting "childname is the name of a child in family Fn" applied to this collection of households on the particular street of families with names F1, F2, . . . F12. Each of the 12 propositions regards whether or not the "argument" childname applies to a child in a particular household. The children's names (childname) can be thought of as the x in a propositional function f(x), where the function is "name of a child in the family with name Fn".  [ original research? ]
Step 1: Assemble all the classes: Whereas the preceding example is finite over the finite propositional function "childnames of the children in family Fn'" on the finite street of a finite number of families, Russell apparently intended the following to extend to all propositional functions extending over an infinite domain so as to allow the creation of all the numbers.
Kleene considers that Russell has set out an impredicative definition that he will have to resolve, or risk deriving something like the Russell paradox. "Here instead we presuppose the totality of all properties of cardinal numbers, as existing in logic, prior to the definition of the natural number sequence" (Kleene 1952:44). The problem will appear, even in the finite example presented here, when Russell deals with the unit class (cf. Russell 1903:517).
The question arises what precisely a "class" is or should be. For Dedekind and Frege, a class is a distinct entity in its own right, a 'unity' that can be identified with all those entities x that satisfy some propositional function F. (This symbolism appears in Russell, attributed there to Frege: "The essence of a function is what is left when the x is taken away, i.e in the above instance, 2( ) 3 + ( ). The argument x does not belong to the function, but the two together make a whole (ib. p. 6 [i.e. Frege's 1891 Function und Begriff]" (Russell 1903:505).) For example, a particular "unity" could be given a name suppose a family Fα has the children with the names Annie, Barbie and Charles:
This notion of collection or class as object, when used without restriction, results in Russell's paradox see more below about impredicative definitions. Russell's solution was to define the notion of a class to be only those elements that satisfy the proposition, his argument being that, indeed, the arguments x do not belong to the propositional function aka "class" created by the function. The class itself is not to be regarded as a unitary object in its own right, it exists only as a kind of useful fiction: "We have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic" (First edition of Principia Mathematica 1927:24).
Russell continues to hold this opinion in his 1919 observe the words "symbolic fictions": [ original research? ]
"When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. . . . But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them . . .." (1919:184)
And in the second edition of PM (1927) Russell holds that "functions occur only through their values, . . . all functions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, lose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.
Step 2: Collect "similar" classes into 'bundles' : These above collections can be put into a "binary relation" (comparing for) similarity by "equinumerosity", symbolized here by ≈, i.e. one-one correspondence of the elements,  and thereby create Russellian classes of classes or what Russell called "bundles". "We can suppose all couples in one bundle, all trios in another, and so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose members are collections, i.e. classes thus each is a class of classes" (Russell 1919:14).
Step 3: Define the null class: Notice that a certain class of classes is special because its classes contain no elements, i.e. no elements satisfy the predicates whose assertion defined this particular class/collection.
The resulting entity may be called "the null class" or "the empty class". Russell symbolized the null/empty class with Λ. So what exactly is the Russellian null class? In PM Russell says that "A class is said to exist when it has at least one member . . . the class which has no members is called the "null class" . . . "α is the null-class" is equivalent to "α does not exist". The question naturally arises whether the null class itself 'exists'? Difficulties related to this question occur in Russell's 1903 work.  After he discovered the paradox in Frege's Grundgesetze he added Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered the need for a "doctrine of types" see more about the unit class, the problem of impredicative definitions and Russell's "vicious circle principle" below. 
Step 4: Assign a "numeral" to each bundle: For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a "numeral"). These symbols are arbitrary.
Step 5: Define "0" Following Frege, Russell picked the empty or null class of classes as the appropriate class to fill this role, this being the class of classes having no members. This null class of classes may be labelled "0"
Step 6: Define the notion of "successor": Russell defined a new characteristic "hereditary" (cf Frege's 'ancestral'), a property of certain classes with the ability to "inherit" a characteristic from another class (which may be a class of classes) i.e. "A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number n, it also belongs to n+1, the successor of n". (1903:21). He asserts that "the natural numbers are the posterity — the "children", the inheritors of the "successor" — of 0 with respect to the relation "the immediate predecessor of (which is the converse of "successor") (1919:23).
Note Russell has used a few words here without definition, in particular "number series", "number n", and "successor". He will define these in due course. Observe in particular that Russell does not use the unit class of classes "1" to construct the successor. The reason is that, in Russell's detailed analysis,  if a unit class becomes an entity in its own right, then it too can be an element in its own proposition this causes the proposition to become "impredicative" and result in a "vicious circle". Rather, he states: "We saw in Chapter II that a cardinal number is to be defined as a class of classes, and in Chapter III that the number 1 is to be defined as the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, unit classes must be defined so as not to assume that we know what is meant by one (1919:181).
For his definition of successor, Russell will use for his "unit" a single entity or "term" as follows:
"It remains to define "successor". Given any number n let α be a class which has n members, and let x be a term which is not a member of α. Then the class consisting of α with x added on will have +1 members. Thus we have the following definition: the successor of the number of terms in the class α is the number of terms in the class consisting of α together with x where x is not any term belonging to the class." (1919:23)
Russell's definition requires a new "term" which is "added into" the collections inside the bundles.
Step 7: Construct the successor of the null class.
Step 8: For every class of equinumerous classes, create its successor.
Step 9: Order the numbers: The process of creating a successor requires the relation " . . . is the successor of . . .", which may be denoted "S", between the various "numerals". "We must now consider the serial character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the class of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later." (1919:31)
Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of "asymmetry" i.e. given the relation such as S (" . . . is the successor of . . . ") between two terms x, and y: x S y ≠ y S x. Second, he defines the notion of "transitivity" for three numerals x, y and z: if x S y and y S z then x S z. Third, he defines the notion of "connected": "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)
He concludes: ". . . [natural] number m is said to be less than another number n when n possesses every hereditary property possessed by the successor of m. It is easy to see, and not difficult to prove, that the relation "less than", so defined, is asymmetrical, transitive, and connected, and has the [natural] numbers for its field [i.e. both domain and converse domain are the numbers]." (1919:35)
The presumption of an 'extralogical' notion of iteration: Kleene notes that "the logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the Intuitionistic view, an essential mathematical kernel is contained in the idea of iteration" (Kleene 1952:46)
Bernays 1930–1931 observes that this notion "two things" already presupposes something, even without the claim of existence of two things, and also without reference to a predicate, which applies to the two things it means, simply, "a thing and one more thing. . . . With respect to this simple definition, the Number concept turns out to be an elementary structural concept . . . the claim of the logicists that mathematics is purely logical knowledge turns out to be blurred and misleading upon closer observation of theoretical logic. . . . [one can extend the definition of "logical"] however, through this definition what is epistemologically essential is concealed, and what is peculiar to mathematics is overlooked" (in Mancosu 1998:243).
Hilbert 1931:266-7, like Bernays, considers there is "something extra-logical" in mathematics: "Besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea of the Kantian epistemology retains its significance: to ascertain the intuitive a priori mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion this is essentially what happens in my investigations of the principles of mathematics. The a priori is here nothing more and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: something is already given to us in advance in our faculty of representation: certain extra-logical concrete objects that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed next to one another is immediately and intuitively given to us, along with the objects, as something that neither can be reduced to anything else, nor needs such a reduction." (Hilbert 1931 in Mancosu 1998: 266, 267).
In brief, according to Hilbert and Bernays, the notion of "sequence" or "successor" is an a priori notion that lies outside symbolic logic.
Hilbert dismissed logicism as a "false path": "Some tried to define the numbers purely logically others simply took the usual number-theoretic modes of inference to be self-evident. On both paths they encountered obstacles that proved to be insuperable." (Hilbert 1931 in Mancoso 1998:267). The incompleteness theorems arguably constitute a similar obstacle for Hilbertian finitism.
Mancosu states that Brouwer concluded that: "the classical laws or principles of logic are part of [the] perceived regularity [in the symbolic representation] they are derived from the post factum record of mathematical constructions . . . Theoretical logic . . . [is] an empirical science and an application of mathematics" (Brouwer quoted by Mancosu 1998:9).
Gödel 1944: With respect to the technical aspects of Russellian logicism as it appears in Principia Mathematica (either edition), Gödel was disappointed:
"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is?] so greatly lacking in formal precision in the foundations (contained in *1–*21 of Principia) that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism" (cf. footnote 1 in Gödel 1944 Collected Works 1990:120).
In particular he pointed out that "The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their definiens" (Russell 1944:120)
With respect to the philosophy that might underlie these foundations, Gödel considered Russell's "no-class theory" as embodying a "nominalistic kind of constructivism . . . which might better be called fictionalism" (cf. footnote 1 in Gödel 1944:119) — to be faulty. See more in "Gödel's criticism and suggestions" below.
Grattan-Guinness: A complicated theory of relations continued to strangle Russell's explanatory 1919 Introduction to Mathematical Philosophy and his 1927 second edition of Principia. Set theory, meanwhile had moved on with its reduction of relation to the ordered pair of sets. Grattan-Guinness observes that in the second edition of Principia Russell ignored this reduction that had been achieved by his own student Norbert Wiener (1914). Perhaps because of "residual annoyance, Russell did not react at all".  By 1914 Hausdorff would provide another, equivalent definition, and Kuratowski in 1921 would provide the one in use today. 
A benign impredicative definition: Suppose a librarian wants to index her collection into a single book (call it Ι for "index"). Her index will list all the books and their locations in the library. As it turns out, there are only three books, and these have titles Ά, β, and Γ. To form her index I, she goes out and buys a book of 200 blank pages and labels it "I". Now she has four books: I, Ά, β, and Γ. Her task is not difficult. When completed, the contents of her index I are 4 pages, each with a unique title and unique location (each entry abbreviated as Title.LocationT):
This sort of definition of I was deemed by Poincaré to be "impredicative". He seems to have considered that only predicative definitions can be allowed in mathematics:
"a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it". 
By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In response to these difficulties, Russell advocated a strong prohibition, his "vicious circle principle":
"No totality can contain members definable only in terms of this totality, or members involving or presupposing this totality" (vicious circle principle)" (Gödel 1944 appearing in Collected Works Vol. II 1990:125). 
A pernicious impredicativity: α = NOT-α: To illustrate what a pernicious instance of impredicativity might be, consider the consequence of inputting argument α into the function f with output ω = 1 – α. This may be seen as the equivalent 'algebraic-logic' expression to the 'symbolic-logic' expression ω = NOT-α, with truth values 1 and 0. When input α = 0, output ω = 1 when input α = 1, output ω = 0.
To make the function "impredicative", identify the input with the output, yielding α = 1-α
Within the algebra of, say, rational numbers the equation is satisfied when α = 0.5. But within, for instance, a Boolean algebra, where only "truth values" 0 and 1 are permitted, then the equality cannot be satisfied.
Fatal impredicativity in the definition of the unit class: Some of the difficulties in the logicist programme may derive from the α = NOT-α paradox  Russell discovered in Frege's 1879 Begriffsschrift  that Frege had allowed a function to derive its input "functional" (value of its variable) not only from an object (thing, term), but also from the function's own output. 
As described above, Both Frege's and Russell's constructions of the natural numbers begin with the formation of equinumerous classes of classes ("bundles"), followed by an assignment of a unique "numeral" to each bundle, and then by the placing of the bundles into an order via a relation S that is asymmetric: x S y ≠ y S x. But Frege, unlike Russell, allowed the class of unit classes to be identified as a unit itself:
But, since the class with numeral 1 is a single object or unit in its own right, it too must be included in the class of unit classes. This inclusion results in an "infinite regress" (as Gödel called it) of increasing "type" and increasing content.
Russell avoided this problem by declaring a class to be more or a "fiction". By this he meant that a class could designate only those elements that satisfied its propositional function and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a "unit". It is an assemblage but is not in Russell's view "worthy of thing-hood":
"The class as many . . . is unobjectionable, but is many and not one. We may, if we choose, represent this by a single symbol: thus x ε u will mean " x is one of the u 's." This must not be taken as a relation of two terms, x and u, because u as the numerical conjunction is not a single term . . . Thus a class of classes will be many many's its constituents will each be only many, and cannot therefore in any sense, one might suppose, be single constituents.[etc]" (1903:516).
This supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem—a hierarchy of "types" of classes.
A solution to impredicativity: a hierarchy of types Edit
Classes as non-objects, as useful fictions: Gödel 1944:131 observes that "Russell adduces two reasons against the extensional view of classes, namely the existence of (1) the null class, which cannot very well be a collection, and (2) the unit classes, which would have to be identical with their single elements." He suggests that Russell should have regarded these as fictitious, but not derive the further conclusion that all classes (such as the class-of-classes that define the numbers 2, 3, etc) are fictions.
But Russell did not do this. After a detailed analysis in Appendix A: The Logical and Arithmetical Doctrines of Frege in his 1903, Russell concludes:
"The logical doctrine which is thus forced upon us is this: The subject of a proposition may be not a single term, but essentially many terms this is the case with all propositions asserting numbers other than 0 and 1" (1903:516).
In the following notice the wording "the class as many"—a class is an aggregate of those terms (things) that satisfy the propositional function, but a class is not a thing-in-itself:
"Thus the final conclusion is, that the correct theory of classes is even more extensional than that of Chapter VI that the class as many is the only object always defined by a propositional function, and that this is adequate for formal purposes" (1903:518).
It is as if a rancher were to round up all his livestock (sheep, cows and horses) into three fictitious corrals (one for the sheep, one for the cows, and one for the horses) that are located in his fictitious ranch. What actually exist are the sheep, the cows and the horses (the extensions), but not the fictitious "concepts" corrals and ranch. [ original research? ]
Ramified theory of types: function-orders and argument-types, predicative functions: When Russell proclaimed all classes are useful fictions he solved the problem of the "unit" class, but the overall problem did not go away rather, it arrived in a new form: "It will now be necessary to distinguish (1) terms, (2) classes, (3) classes of classes, and so on ad infinitum we shall have to hold that no member of one set is a member of any other set, and that x ε u requires that x should be of a set of a degree lower by one than the set to which u belongs. Thus x ε x will become a meaningless proposition and in this way the contradiction is avoided" (1903:517).
This is Russell's "doctrine of types". To guarantee that impredicative expressions such as x ε x can be treated in his logic, Russell proposed, as a kind of working hypothesis, that all such impredicative definitions have predicative definitions. This supposition requires the notions of function-"orders" and argument-"types". First, functions (and their classes-as-extensions, i.e. "matrices") are to be classified by their "order", where functions of individuals are of order 1, functions of functions (classes of classes) are of order 2, and so forth. Next, he defines the "type" of a function's arguments (the function's "inputs") to be their "range of significance", i.e. what are those inputs α (individuals? classes? classes-of-classes? etc.) that, when plugged into f(x), yield a meaningful output ω. Note that this means that a "type" can be of mixed order, as the following example shows:
"Joe DiMaggio and the Yankees won the 1947 World Series".
This sentence can be decomposed into two clauses: "x won the 1947 World Series" + "y won the 1947 World Series". The first sentence takes for x an individual "Joe DiMaggio" as its input, the other takes for y an aggregate "Yankees" as its input. Thus the composite-sentence has a (mixed) type of 2, mixed as to order (1 and 2).
By "predicative", Russell meant that the function must be of an order higher than the "type" of its variable(s). Thus a function (of order 2) that creates a class of classes can only entertain arguments for its variable(s) that are classes (type 1) and individuals (type 0), as these are lower types. Type 3 can only entertain types 2, 1 or 0, and so forth. But these types can be mixed (for example, for this sentence to be (sort of) true: " z won the 1947 World Series " could accept the individual (type 0) "Joe DiMaggio" and/or the names of his other teammates, and it could accept the class (type 1) of individual players "The Yankees".
The axiom of reducibility: The axiom of reducibility is the hypothesis that any function of any order can be reduced to (or replaced by) an equivalent predicative function of the appropriate order.  A careful reading of the first edition indicates that an n th order predicative function need not be expressed "all the way down" as a huge "matrix" or aggregate of individual atomic propositions. "For in practice only the relative types of variables are relevant thus the lowest type occurring in a given context may be called that of individuals" (p. 161). But the axiom of reducibility proposes that in theory a reduction "all the way down" is possible.
Russell 1927 abandons the axiom of reducibility: By the 2nd edition of PM of 1927, though, Russell had given up on the axiom of reducibility and concluded he would indeed force any order of function "all the way down" to its elementary propositions, linked together with logical operators:
"All propositions, of whatever order, are derived from a matrix composed of elementary propositions combined by means of the stroke" (PM 1927 Appendix A, p. 385)
(The "stroke" is Sheffer's stroke — adopted for the 2nd edition of PM — a single two argument logical function from which all other logical functions may be defined.)
The net result, though, was a collapse of his theory. Russell arrived at this disheartening conclusion: that "the theory of ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealt with. . . . Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom" (PM 1927:xiv).
Gödel 1944 agrees that Russell's logicist project was stymied he seems to disagree that even the integers survived:
"[In the second edition] The axiom of reducibility is dropped, and it is stated explicitly that all primitive predicates belong to the lowest type and that the only purpose of variables (and evidently also of constants) of higher orders and types is to make it possible to assert more complicated truth-functions of atomic propositions" (Gödel 1944 in Collected Works:134).
Gödel asserts, however, that this procedure seems to presuppose arithmetic in some form or other (p. 134). He deduces that "one obtains integers of different orders" (p. 134-135) the proof in Russell 1927 PM Appendix B that "the integers of any order higher than 5 are the same as those of order 5" is "not conclusive" and "the question whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy [classes plus types] must be considered as unsolved at the present time". Gödel concluded that it wouldn't matter anyway because propositional functions of order n (any n) must be described by finite combinations of symbols (all quotes and content derived from page 135).
Gödel's criticism and suggestions Edit
Gödel, in his 1944 work, identifies the place where he considers Russell's logicism to fail and offers suggestions to rectify the problems. He submits the "vicious circle principle" to re-examination, splitting it into three parts "definable only in terms of", "involving" and "presupposing". It is the first part that "makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of mathematics itself". Since, he argues, mathematics sees to rely on its inherent impredicativities (e.g. "real numbers defined by reference to all real numbers"), he concludes that what he has offered is "a proof that the vicious circle principle is false [rather] than that classical mathematics is false" (all quotes Gödel 1944:127).
Russell's no-class theory is the root of the problem: Gödel believes that impredicativity is not "absurd", as it appears throughout mathematics. Russell's problem derives from his "constructivistic (or nominalistic"  ) standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . a notion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction" (p. 128).
Indeed, Russell's "no class" theory, Gödel concludes:
"is of great interest as one of the few examples, carried out in detail, of the tendency to eliminate assumptions about the existence of objects outside the "data" and to replace them by constructions on the basis of these data 33 . The "data" are to understand in a relative sense here i.e. in our case as logic without the assumption of the existence of classes and concepts]. The result has been in this case essentially negative i.e. the classes and concepts introduced in this way do not have all the properties required from their use in mathematics. . . . All this is only a verification of the view defended above that logic and mathematics (just as physics) are built up on axioms with a real content which cannot be explained away" (p. 132)
He concludes his essay with the following suggestions and observations:
"One should take a more conservative course, such as would consist in trying to make the meaning of terms "class" and "concept" clearer, and to set up a consistent theory of classes and concepts as objectively existing entities. This is the course which the actual development of mathematical logic has been taking and which Russell himself has been forced to enter upon in the more constructive parts of his work. Major among the attempts in this direction . . . are the simple theory of types . . . and axiomatic set theory, both of which have been successful at least to this extent, that they permit the derivation of modern mathematics and at the same time avoid all known paradoxes . . . ¶ It seems reasonable to suspect that it is this incomplete understanding of the foundations which is responsible for the fact that mathematical logic has up to now remained so far behind the high expectations of Peano and others . . .." (p. 140)
Neo-logicism describes a range of views considered by their proponents to be successors of the original logicist program.  More narrowly, neo-logicism may be seen as the attempt to salvage some or all elements of Frege's programme through the use of a modified version of Frege's system in the Grundgesetze (which may be seen as a kind of second-order logic).
For instance, one might replace Basic Law V (analogous to the axiom schema of unrestricted comprehension in naive set theory) with some 'safer' axiom so as to prevent the derivation of the known paradoxes. The most cited candidate to replace BLV is Hume's principle, the contextual definition of '#' given by '#F = #G if and only if there is a bijection between F and G'.  This kind of neo-logicism is often referred to as neo-Fregeanism.  Proponents of neo-Fregeanism include Crispin Wright and Bob Hale, sometimes also called the Scottish School or abstractionist Platonism,  who espouse a form of epistemic foundationalism. 
Other major proponents of neo-logicism include Bernard Linsky and Edward N. Zalta, sometimes called the Stanford–Edmonton School, abstract structuralism or modal neo-logicism who espouse a form of axiomatic metaphysics.   Modal neo-logicism derives the Peano axioms within second-order modal object theory.  
Another quasi-neo-logicist approach has been suggested by M. Randall Holmes. In this kind of amendment to the Grundgesetze, BLV remains intact, save for a restriction to stratifiable formulae in the manner of Quine's NF and related systems. Essentially all of the Grundgesetze then 'goes through'. The resulting system has the same consistency strength as Jensen's NFU + Rosser's Axiom of Counting. 
I will be writing out my solutions to problems in Algebraic Curves, by Fulton, which is an undergraduate introduction to algebraic geometry. I will also be including summaries of each section. The text can be obtained as a PDF by a simple Google search (via Fulton himself). Throughout this venture, many solutions will be sketched, and some may be wrong in various places. Feedback is welcomed. Note: all rings in this text will be assumed to commutative, with identity.
Section 1.1 Recap: The first section is primarily a review of basic algebraic ideas. One should be familiar with (commutative) rings, including domains, fields, UFDs, PIDs, and quotient rings. One should also be familiar with types of ideals (prime, maximal, principal). The real algebraic geometry begins in section 1.2. The exercises for section 1.1 are also all algebra, so I omitted most.
1.3 Let be a PID, and let be a nonzero, proper, prime ideal in . Then
(a) is generated by an irreducible element, and
(b) is maximal.
Proof. (a) Since is a PID, write . First, is not a unit otherwise . Also, is not by assumption. Suppose , where neither nor is a unit. Then either or is in since is prime. Suppose WLOG that , so that . Then , which implies that since is a domain. But is not a unit and thus has no multiplicative inverse, a contradiction.
(b) Let , and suppose , for some ideal . Since , we have . Since , we also have . Furthermore, since is prime, either or . By assumption, (otherwise ). Hence . But is not a unit since , and so must be a unit since is irreducible by (a). This implies that , a contradiction. Hence must be maximal.
1.4 Let be an infinite field, and let . Suppose for all . Then .
Proof. We show that if , then there exists some such that . The proof is by induction on . Suppose and . Then the degree of is at least the number of roots of . Since the degree is finite, the number of roots is finite. Since is infinite, there are elements in that are not a root of .
Now, fix and assume the claim for . Write , where . Since , there exists some such that . By the induction hypothesis, there exists such that . Hence is a nonzero polynomial in (recall that the coefficients of are grouped together into so we cannot have a situation like being zero on ). By the case (since is infinite), there must be some such that , which is what we wanted.
Coursework Policy "3+2+2" Requirement
The department offers two-semester “Prelim” course sequences in six core areas, making twelve Prelim segments in total: Algebra Analysis (Real and Complex) Applied Mathematics (principally functional analysis) Numerical Analysis Probability and Topology (Algebraic and Differential). The course syllabi can be found here.
90-minute examinations (“prelim exams”) covering the twelve areas are administered twice per academic year: in August before the start of the Fall semester, and in January before the start of the Spring semester.
Whenever possible, exams covering different areas are administered on different days. The two exams covering a single area (for instance, the two Algebra exams) are administered sequentially on the same day, with a brief rest break in between.
Of the 12 Prelim segments, students must pass at least 7, in distinct areas, of which at least 3 must be by exam. A passing grade in a Prelim course is a “B”, while the passing standard for a Prelim exam is determined by the faculty committee administering that exam.
Students are expected to meet the following milestones:
By the beginning of their 2nd Semester: Pass 1 Prelim exam.
“3+2”: By the beginning of their 4th Semester: pass 5 Prelim segments, all distinct, at least 3 by exam. Note that you become eligible to take a candidacy oral exam once the above requirements are completed.
“3+2+2”: By the beginning of their 8th semester (or before the beginning of the semester of their Ph.D. thesis defense, whichever comes first): pass two additional prelim courses or exams, distinct from the “3+2”.
While this schedule is not rigidly enforced, students are expected to make steady progress towards the completion of their prelim requirements. Students falling distinctly behind this schedule risk losing their good academic standing within the program, and should consult the Graduate Advisor.
Students are welcome to take both a prelim course and the corresponding exam, though they cannot both be counted towards the prelim requirements. There is no penalty for failing a prelim exam they can be retaken on subsequent occasions. There are no waivers for prelim exam requirements.
There are three ways in which students may be allowed to skip some or all of required prelim courses:
As implied by the above rules, they may pass more exams instead of taking the courses.
Students with prior graduate coursework may appeal to the Graduate Advisor for waivers of one or more prelim course requirements.
Interdisciplinary students, with the advice of an academic supervisor and permission of the ASGSC (Administrative Subcommittee of the Graduate Studies Committee) may be allowed to substitute courses in their specialty for some of the four required prelim courses.
DEPARTMENT OF MATHEMATICS
All students must pass the Preliminary Examination in order to continue with the program. The Preliminary Examination consists of written examinations in the following three subjects: algebra, analysis, and geometry/topology. Incoming students are strongly encouraged to take the Preliminary Examination upon entrance. There is no penalty for failing to pass a preliminary examination taken upon entrance to the program.Incoming students are required to take each of the first-year courses (Algebra, Analysis, and Geometry-Topology) for which they do not pass the preliminary exam.
The Preliminary Examination is given during New Student Week in September and at the end of the academic year (typically June). Graduate students must take the Preliminary Examination in all three subjects by the end of their first academic year. Students who do not pass the Preliminary Examination by the end of their first year must pass a make-up examination in September of their second year in order to continue in the program beyond the first quarter of the second year.
An award is offered at the end of each academic year to the student who has achieved the best performance in the Preliminary Examination. The award is accompanied by a monetary prize when sufficient funds are available.
Students who do not pass the Preliminary Examination by the end of their first year must pass a make-up examination in September of their second year in order to continue in the program beyond the first quarter of the second year. This rule may be waived with the permission of the instructor and the approval of the Graduate Committee.
To help the graduate students (including incoming and prospective students) better prepare for the exams, we provide here some sets of problems from past years. Trying to solve these problems is an excellent way to find out in which areas more work is needed. We encourage prospective graduate students to study books and audit courses related to the material of these exams while still in college.
Analysis (Real and Complex) (410)
Geometry and Topology (Pre-2009)
Prior to Fall 2009, Topology and Geometry were courses with separate examinations. These examples should be used only as a source of problems, not as guidelines to the syllabus for or the structure of current exams. The current syllabus is below under Geometry/Topology (440).
3. The Later Wittgenstein on Mathematics: Some Preliminaries
The first and most important thing to note about Wittgenstein&rsquos later Philosophy of Mathematics is that RFM, first published in 1956, consists of selections taken from a number of manuscripts (1937&ndash1944), most of one large typescript (1938), and three short typescripts (1938), each of which constitutes an Appendix to (RFM I). For this reason and because some manuscripts containing much material on mathematics (e.g., MS 123) were not used at all for RFM, philosophers have not been able to read Wittgenstein&rsquos later remarks on mathematics as they were written in the manscripts used for RFM and they have not had access (until the 2000&ndash2001 release of the Nachlass on CD-ROM) to much of Wittgenstein&rsquos later work on mathematics. It must be emphasized, therefore, that this Encyclopedia article is being written during a transitional period. Until philosophers have used the Nachlass to build a comprehensive picture of Wittgenstein&rsquos complete and evolving Philosophy of Mathematics, we will not be able to say definitively which views the later Wittgenstein retained, which he changed, and which he dropped. In the interim, this article will outline Wittgenstein&rsquos later Philosophy of Mathematics, drawing primarily on RFM, to a much lesser extent LFM (1939 Cambridge lectures), and, where possible, previously unpublished material in Wittgenstein&rsquos Nachlass.
It should also be noted at the outset that commentators disagree about the continuity of Wittgenstein&rsquos middle and later Philosophies of Mathematics. Some argue that the later views are significantly different from the intermediate views (Frascolla 1994 Gerrard 1991: 127, 131&ndash32 Floyd 2005: 105&ndash106), while others argue that, for the most part, Wittgenstein&rsquos Philosophy of Mathematics evolves from the middle to the later period without significant changes or renunciations (Wrigley 1993 Marion 1998). The remainder of this article adopts the second interpretation, explicating Wittgenstein&rsquos later Philosophy of Mathematics as largely continuous with his intermediate views, except for the important introduction of an extra-mathematical application criterion.
3.1 Mathematics as a Human Invention
Perhaps the most important constant in Wittgenstein&rsquos Philosophy of Mathematics, middle and late, is that he consistently maintains that mathematics is our, human invention, and that, indeed, everything in mathematics is invented. Just as the middle Wittgenstein says that &ldquo[w]e make mathematics&rdquo, the later Wittgenstein says that we &lsquoinvent&rsquo mathematics (RFM I, §168 II, §38 V, §§5, 9 and 11 PG 469&ndash70) and that &ldquothe mathematician is not a discoverer: he is an inventor&rdquo (RFM, Appendix II, §2 (LFM 22, 82). Nothing exists mathematically unless and until we have invented it.
In arguing against mathematical discovery, Wittgenstein is not just rejecting Platonism, he is also rejecting a rather standard philosophical view according to which human beings invent mathematical calculi, but once a calculus has been invented, we thereafter discover finitely many of its infinitely many provable and true theorems. As Wittgenstein himself asks (RFM IV, §48), &ldquomight it not be said that the rules lead this way, even if no one went it?&rdquo If &ldquosomeone produced a proof [of &lsquoGoldbach&rsquos theorem&rsquo]&rdquo, &ldquo[c]ouldn&rsquot one say&rdquo, Wittgenstein asks (LFM 144), &ldquothat the possibility of this proof was a fact in the realms of mathematical reality&rdquo&mdashthat &ldquo[i]n order [to] find it, it must in some sense be there&rdquo&mdash&ldquo[i]t must be a possible structure&rdquo?
Unlike many or most philosophers of mathematics, Wittgenstein resists the &lsquoYes&rsquo answer that we discover truths about a mathematical calculus that come into existence the moment we invent the calculus (PR §141 PG 283, 466 LFM 139). Wittgenstein rejects the modal reification of possibility as actuality&mdashthat provability and constructibility are (actual) facts&mdashby arguing that it is at the very least wrong-headed to say with the Platonist that because &ldquoa straight line can be drawn between any two points,&hellip the line already exists even if no one has drawn it&rdquo&mdashto say &ldquo[w]hat in the ordinary world we call a possibility is in the geometrical world a reality&rdquo (LFM 144 RFM I, §21). One might as well say, Wittgenstein suggests (PG 374), that &ldquochess only had to be discovered, it was always there!&rdquo
At MS 122 (3v Oct. 18, 1939), Wittgenstein once again emphasizes the difference between illusory mathematical discovery and genuine mathematical invention.
I want to get away from the formulation: &ldquoI now know more about the calculus&rdquo, and replace it with &ldquoI now have a different calculus&rdquo. The sense of this is always to keep before one&rsquos eyes the full scale of the gulf between a mathematical knowing and non-mathematical knowing. 
And as with the middle period, the later Wittgenstein similarly says (MS 121, 27r May 27, 1938) that &ldquo[i]t helps if one says: the proof of the Fermat proposition is not to be discovered, but to be invented&rdquo.
The difference between the &lsquoanthropological&rsquo and the mathematical account is that in the first we are not tempted to speak of &lsquomathematical facts&rsquo, but rather that in this account the facts are never mathematical ones, never make mathematical propositions true or false. (MS 117, 263 March 15, 1940)
There are no mathematical facts just as there are no (genuine) mathematical propositions. Repeating his intermediate view, the later Wittgenstein says (MS 121, 71v 27 Dec., 1938): &ldquoMathematics consists of [calculi | calculations], not of propositions&rdquo. This radical constructivist conception of mathematics prompts Wittgenstein to make notorious remarks&mdashremarks that virtually no one else would make&mdashsuch as the infamous (RFM V, §9): &ldquoHowever queer it sounds, the further expansion of an irrational number is a further expansion of mathematics&rdquo.
3.1.1 Wittgenstein&rsquos Later Anti-Platonism: The Natural History of Numbers and the Vacuity of Platonism
As in the middle period, the later Wittgenstein maintains that mathematics is essentially syntactical and non-referential, which, in and of itself, makes Wittgenstein&rsquos philosophy of mathematics anti-Platonist insofar as Platonism is the view that mathematical terms and propositions refer to objects and/or facts and that mathematical propositions are true by virtue of agreeing with mathematical facts.
The later Wittgenstein, however, wishes to &lsquowarn&rsquo us that our thinking is saturated with the idea of &ldquo[a]rithmetic as the natural history (mineralogy) of numbers&rdquo (RFM IV, §11). When, for instance, Wittgenstein discusses the claim that fractions cannot be ordered by magnitude, he says that this sounds &lsquoremarkable&rsquo in a way that a mundane proposition of the differential calculus does not, for the latter proposition is associated with an application in physics,
whereas this proposition &hellip seems to [solely] concern&hellip the natural history of mathematical objects themselves. (RFM II, §40)
Wittgenstein stresses that he is trying to &lsquowarn&rsquo us against this &lsquoaspect&rsquo&mdashthe idea that the foregoing proposition about fractions &ldquointroduces us to the mysteries of the mathematical world&rdquo, which exists somewhere as a completed totality, awaiting our prodding and our discoveries. The fact that we regard mathematical propositions as being about mathematical objects and mathematical investigation &ldquoas the exploration of these objects&rdquo is &ldquoalready mathematical alchemy&rdquo, claims Wittgenstein (RFM V, §16), since
it is not possible to appeal to the meaning [Bedeutung] of the signs in mathematics,&hellip because it is only mathematics that gives them their meaning [Bedeutung].
Platonism is dangerously misleading, according to Wittgenstein, because it suggests a picture of pre-existence, predetermination and discovery that is completely at odds with what we find if we actually examine and describe mathematics and mathematical activity. &ldquoI should like to be able to describe&rdquo, says Wittgenstein (RFM IV, §13), &ldquohow it comes about that mathematics appears to us now as the natural history of the domain of numbers, now again as a collection of rules&rdquo.
Wittgenstein, however, does not endeavour to refute Platonism. His aim, instead, is to clarify what Platonism is and what it says, implicitly and explicitly (including variants of Platonism that claim, e.g., that if a proposition is provable in an axiom system, then there already exists a path [i.e., a proof] from the axioms to that proposition (RFM I, §21 Marion 1998: 13&ndash14, 226 Steiner 2000: 334). Platonism is either &ldquoa mere truism&rdquo (LFM 239), Wittgenstein says, or it is a &lsquopicture&rsquo consisting of &ldquoan infinity of shadowy worlds&rdquo (LFM 145), which, as such, lacks &lsquoutility&rsquo (cf. PI §254) because it explains nothing and it misleads at every turn.
3.2 Wittgenstein&rsquos Later Finitistic Constructivism
Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986: 300&ndash301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite (RFM V, §21 Zettel §274, 1947) and infinite mathematical extensions.
The first, and perhaps most definitive, indication that the later Wittgenstein maintains his finitism is his continued and consistent insistence that irrational numbers are rules for constructing finite expansions, not infinite mathematical extensions. &ldquoThe concepts of infinite decimals in mathematical propositions are not concepts of series&rdquo, says Wittgenstein (RFM V, §19), &ldquobut of the unlimited technique of expansion of series&rdquo. We are misled by &ldquo[t]he extensional definitions of functions, of real numbers etc&rdquo. (RFM V, §35), but once we recognize the Dedekind cut as &ldquoan extensional image&rdquo, we see that we are not &ldquoled to (sqrt<2>) by way of the concept of a cut&rdquo (RFM V, §34). On the later Wittgenstein&rsquos account, there simply is no property, no rule, no systematic means of defining each and every irrational number intensionally, which means there is no criterion &ldquofor the irrational numbers being complete&rdquo (PR §181).
As in his intermediate position, the later Wittgenstein claims that &lsquo(aleph_0)&rsquo and &ldquoinfinite series&rdquo get their mathematical uses from the use of &lsquoinfinity&rsquo in ordinary language (RFM II, §60). Although, in ordinary language, we often use &lsquoinfinite&rsquo and &ldquoinfinitely many&rdquo as answers to the question &ldquohow many?&rdquo, and though we associate infinity with the enormously large, the principal use we make of &lsquoinfinite&rsquo and &lsquoinfinity&rsquo is to speak of the unlimited (RFM V, §14) and unlimited techniques (RFM II, §45 PI §218). This fact is brought out by the fact &ldquothat the technique of learning (aleph_0) numerals is different from the technique of learning 100,000 numerals&rdquo (LFM 31). When we say, e.g., that &ldquothere are an infinite number of even numbers&rdquo we mean that we have a mathematical technique or rule for generating even numbers which is limitless, which is markedly different from a limited technique or rule for generating a finite number of numbers, such as 1&ndash100,000,000. &ldquoWe learn an endless technique&rdquo, says Wittgenstein (RFM V, §19), &ldquobut what is in question here is not some gigantic extension&rdquo.
An infinite sequence, for example, is not a gigantic extension because it is not an extension, and &lsquo(aleph_0)&rsquo is not a cardinal number, for &ldquohow is this picture connected with the calculus&rdquo, given that &ldquoits connexion is not that of the picture | | | | with 4&rdquo (i.e., given that &lsquo(aleph_0)&rsquo is not connected to a (finite) extension)? This shows, says Wittgenstein (RFM II, §58), that we ought to avoid the word &lsquoinfinite&rsquo in mathematics wherever it seems to give a meaning to the calculus, rather than acquiring its meaning from the calculus and its use in the calculus. Once we see that the calculus contains nothing infinite, we should not be &lsquodisappointed&rsquo (RFM II, §60), but simply note (RFM II, §59) that it is not &ldquoreally necessary&hellip to conjure up the picture of the infinite (of the enormously big)&rdquo.
A second strong indication that the later Wittgenstein maintains his finitism is his continued and consistent treatment of &lsquopropositions&rsquo of the type &ldquoThere are three consecutive 7s in the decimal expansion of (pi)&rdquo (hereafter &lsquoPIC&rsquo).  In the middle period, PIC (and its putative negation, ( eg)PIC, namely, &ldquoIt is not the case that there are three consecutive 7s in the decimal expansion of (pi)&rdquo) is not a meaningful mathematical &ldquostatement at all&rdquo (WVC 81&ndash82: note 1). On Wittgenstein&rsquos intermediate view, PIC&mdashlike FLT, GC, and the Fundamental Theorem of Algebra&mdashis not a mathematical proposition because we do not have in hand an applicable decision procedure by which we can decide it in a particular calculus. For this reason, we can only meaningfully state finitistic propositions regarding the expansion of (pi), such as &ldquoThere exist three consecutive 7s in the first 10,000 places of the expansion of (pi)&rdquo (WVC 71 81&ndash82, note 1).
The later Wittgenstein maintains this position in various passages in RFM (Bernays 1959: 11&ndash12). For example, to someone who says that since &ldquothe rule of expansion determine[(s)] the series completely&rdquo, &ldquoit must implicitly determine all questions about the structure of the series&rdquo, Wittgenstein replies: &ldquoHere you are thinking of finite series&rdquo (RFM V, §11). If PIC were a mathematical question (or problem)&mdashif it were finitistically restricted&mdashit would be algorithmically decidable, which it is not (RFM V, §21 LFM 31&ndash32, 111, 170 WVC 102&ndash03). As Wittgenstein says at (RFM V, §9): &ldquoThe question&hellip changes its status, when it becomes decidable&rdquo, &ldquo[f]or a connexion is made then, which formerly was not there&rdquo. And if, moreover, one invokes the Law of the Excluded Middle to establish that PIC is a mathematical proposition&mdashi.e., by saying that one of these &ldquotwo pictures&hellip must correspond to the fact&rdquo (RFM V, §10)&mdashone simply begs the question (RFM V, §12), for if we have doubts about the mathematical status of PIC, we will not be swayed by a person who asserts &ldquoPIC (vee eg)PIC&rdquo (RFM VII, §41 V, §13).
Wittgenstein&rsquos finitism, constructivism, and conception of mathematical decidability are interestingly connected at (RFM VII, §41, par. 2&ndash5).
What harm is done e.g. by saying that God knows all irrational numbers? Or: that they are already there, even though we only know certain of them? Why are these pictures not harmless?
For one thing, they hide certain problems.&mdash (MS 124: 139 March 16, 1944)
Suppose that people go on and on calculating the expansion of (pi). So God, who knows everything, knows whether they will have reached &lsquo777&rsquo by the end of the world. But can his omniscience decide whether they would have reached it after the end of the world? It cannot. I want to say: Even God can determine something mathematical only by mathematics. Even for him the mere rule of expansion cannot decide anything that it does not decide for us.
We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is a &lsquo2&rsquo at the fifth place. Could God have known this, without the calculation, purely from the rule of expansion? I want to say: No. (MS 124, pp. 175&ndash176 March 23&ndash24, 1944)
What Wittgenstein means here is that God&rsquos omniscience might, by calculation, find that &lsquo777&rsquo occurs at the interval [(n,n+2)], but, on the other hand, God might go on calculating forever without &lsquo777&rsquo ever turning up. Since (pi) is not a completed infinite extension that can be completely surveyed by an omniscient being (i.e., it is not a fact that can be known by an omniscient mind), even God has only the rule, and so God&rsquos omniscience is no advantage in this case (LFM 103&ndash04 cf. Weyl 1921 [1998: 97]). Like us, with our modest minds, an omniscient mind (i.e., God) can only calculate the expansion of (pi) to some (n) th decimal place&mdashwhere our (n) is minute and God&rsquos (n) is (relatively) enormous&mdashand at no (n) th decimal place could any mind rightly conclude that because &lsquo777&rsquo has not turned up, it, therefore, will never turn up.
3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability
On one fairly standard interpretation, the later Wittgenstein says that &ldquotrue in calculus (Gamma)&rdquo is identical to &ldquoprovable in calculus (Gamma)&rdquo and, therefore, that a mathematical proposition of calculus (Gamma) is a concatenation of signs that is either provable (in principle) or refutable (in principle) in calculus (Gamma) (Goodstein 1972: 279, 282 Anderson 1958: 487 Klenk 1976: 13 Frascolla 1994: 59). On this interpretation, the later Wittgenstein precludes undecidable mathematical propositions, but he allows that some undecided expressions are propositions of a calculus because they are decidable in principle (i.e., in the absence of a known, applicable decision procedure).
There is considerable evidence, however, that the later Wittgenstein maintains his intermediate position that an expression is a meaningful mathematical proposition only within a given calculus and iff we knowingly have in hand an applicable and effective decision procedure by means of which we can decide it. For example, though Wittgenstein vacillates between &ldquoprovable in PM&rdquo and &ldquoproved in PM&rdquo at (RFM App. III, §6, §8), he does so in order to use the former to consider the alleged conclusion of Gödel&rsquos proof (i.e., that there exist true but unprovable mathematical propositions), which he then rebuts with his own identification of &ldquotrue in calculus (Gamma)&rdquo with &ldquoproved in calculus (Gamma)&rdquo (i.e., not with &ldquoprovable in calculus (Gamma)&rdquo) (Wang 1991: 253 Rodych 1999a: 177). This construal is corroborated by numerous passages in which Wittgenstein rejects the received view that a provable but unproved proposition is true, as he does when he asserts that (RFM III, §31, 1939) a proof &ldquomakes new connexions&rdquo, &ldquo[i]t does not establish that they are there&rdquo because &ldquothey do not exist until it makes them&rdquo, and when he says (RFM VII, §10, 1941) that &ldquo[a] new proof gives the proposition a place in a new system&rdquo. Furthermore, as we have just seen, Wittgenstein rejects PIC as a non-proposition on the grounds that it is not algorithmically decidable, while admitting finitistic versions of PIC because they are algorithmically decidable.
Perhaps the most compelling evidence that the later Wittgenstein maintains algorithmic decidability as his criterion for a mathematical proposition lies in the fact that, at (RFM V, §9, 1942), he says in two distinct ways that a mathematical &lsquoquestion&rsquo can become decidable and that when this happens, a new connexion is &lsquomade&rsquo which previously did not exist. Indeed, Wittgenstein cautions us against appearances by saying that &ldquoit looks as if a ground for the decision were already there&rdquo, when, in fact, &ldquoit has yet to be invented&rdquo. These passages strongly militate against the claim that the later Wittgenstein grants that proposition (phi) is decidable in calculus (Gamma) iff it is provable or refutable in principle. Moreover, if Wittgenstein held this position, he would claim, contra (RFM V, §9), that a question or proposition does not become decidable since it simply (always) is decidable. If it is provable, and we simply don&rsquot yet know this to be the case, there already is a connection between, say, our axioms and rules and the proposition in question. What Wittgenstein says, however, is that the modalities provable and refutable are shadowy forms of reality&mdashthat possibility is not actuality in mathematics (PR §§141, 144, 172 PG 281, 283, 299, 371, 466, 469 LFM 139). Thus, the later Wittgenstein agrees with the intermediate Wittgenstein that the only sense in which an undecided mathematical proposition (RFM VII, §40, 1944) can be decidable is in the sense that we know how to decide it by means of an applicable decision procedure.
3.4 Wittgenstein&rsquos Later Critique of Set Theory: Non-Enumerability vs. Non-Denumerability
Largely a product of his anti-foundationalism and his criticism of the extension-intension conflation, Wittgenstein&rsquos later critique of set theory is highly consonant with his intermediate critique (PR §§109, 168 PG 334, 369, 469 LFM 172, 224, 229 and RFM III, §43, 46, 85, 90 VII, §16). Given that mathematics is a &ldquoMOTLEY of techniques of proof&rdquo (RFM III, §46), it does not require a foundation (RFM VII, §16) and it cannot be given a self-evident foundation (PR §160 WVC 34 & 62 RFM IV, §3). Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary.
Even if set theory is unnecessary, it still might constitute a solid foundation for mathematics. In his core criticism of set theory, however, the later Wittgenstein denies this, saying that the diagonal proof does not prove non-denumerability, for &ldquo[i]t means nothing to say: &lsquoTherefore the X numbers are not denumerable&rsquo&rdquo (RFM II, §10). When the diagonal is construed as a proof of greater and lesser infinite sets it is a &ldquopuffed-up proof&rdquo, which, as Poincaré argued (1913: 61&ndash62), purports to prove or show more than &ldquoits means allow it&rdquo (RFM II, §21).
If it were said: Consideration of the diagonal procedure shews you that the concept &lsquoreal number&rsquo has much less analogy with the concept &lsquocardinal number&rsquo than we, being misled by certain analogies, are inclined to believe, that would have a good and honest sense. But just the opposite happens: one pretends to compare the &lsquoset&rsquo of real numbers in magnitude with that of cardinal numbers. The difference in kind between the two conceptions is represented, by a skew form of expression, as difference of extension. I believe, and hope, that a future generation will laugh at this hocus pocus. (RFM II, §22)
The sickness of a time is cured by an alteration in the mode of life of human beings&hellip (RFM II, §23)
The &ldquohocus pocus&rdquo of the diagonal proof rests, as always for Wittgenstein, on a conflation of extension and intension, on the failure to properly distinguish sets as rules for generating extensions and (finite) extensions. By way of this confusion &ldquoa difference in kind&rdquo (i.e., unlimited rule vs. finite extension) &ldquois represented by a skew form of expression&rdquo, namely as a difference in the cardinality of two infinite extensions. Not only can the diagonal not prove that one infinite set is greater in cardinality than another infinite set, according to Wittgenstein, nothing could prove this, simply because &ldquoinfinite sets&rdquo are not extensions, and hence not infinite extensions. But instead of interpreting Cantor&rsquos diagonal proof honestly, we take the proof to &ldquoshow there are numbers bigger than the infinite&rdquo, which &ldquosets the whole mind in a whirl, and gives the pleasant feeling of paradox&rdquo (LFM 16&ndash17)&mdasha &ldquogiddiness attacks us when we think of certain theorems in set theory&rdquo&mdash&ldquowhen we are performing a piece of logical sleight-of-hand&rdquo (PI §412 §426 1945). This giddiness and pleasant feeling of paradox, says Wittgenstein (LFM 16), &ldquomay be the chief reason [set theory] was invented&rdquo.
Though Cantor&rsquos diagonal is not a proof of non-denumerability, when it is expressed in a constructive manner, as Wittgenstein himself expresses it at (RFM II, §1), &ldquoit gives sense to the mathematical proposition that the number so-and-so is different from all those of the system&rdquo (RFM II, §29). That is, the proof proves non-enumerability: it proves that for any given definite real number concept (e.g., recursive real), one cannot enumerate &lsquoall&rsquo such numbers because one can always construct a diagonal number, which falls under the same concept and is not in the enumeration. &ldquoOne might say&rdquo, Wittgenstein says,
I call number-concept X non-denumerable if it has been stipulated that, whatever numbers falling under this concept you arrange in a series, the diagonal number of this series is also to fall under that concept. (RFM II, §10 cf. II, §§30, 31, 13)
One lesson to be learned from this, according to Wittgenstein (RFM II, §33), is that &ldquothere are diverse systems of irrational points to be found in the number line&rdquo, each of which can be given by a recursive rule, but &ldquono system of irrational numbers&rdquo, and &ldquoalso no super-system, no &lsquoset of irrational numbers&rsquo of higher-order infinity&rdquo. Cantor has shown that we can construct &ldquoinfinitely many&rdquo diverse systems of irrational numbers, but we cannot construct an exhaustive system of all the irrational numbers (RFM II, §29). As Wittgenstein says at (MS 121, 71r Dec. 27, 1938), three pages after the passage used for (RFM II, §57):
If you now call the Cantorian procedure one for producing a new real number, you will now no longer be inclined to speak of a system of all real numbers. (italics added)
From Cantor&rsquos proof, however, set theorists erroneously conclude that &ldquothe set of irrational numbers&rdquo is greater in multiplicity than any enumeration of irrationals (or the set of rationals), when the only conclusion to draw is that there is no such thing as the set of all the irrational numbers. The truly dangerous aspect to &lsquopropositions&rsquo such as &ldquoThe real numbers cannot be arranged in a series&rdquo and &ldquoThe set&hellip is not denumerable&rdquo is that they make concept formation [i.e., our invention] &ldquolook like a fact of nature&rdquo (i.e., something we discover) (RFM II §§16, 37). At best, we have a vague idea of the concept of &ldquoreal number&rdquo, but only if we restrict this idea to &ldquorecursive real number&rdquo and only if we recognize that having the concept does not mean having a set of all recursive real numbers.
3.5 Extra-Mathematical Application as a Necessary Condition of Mathematical Meaningfulness
The principal and most significant change from the middle to later writings on mathematics is Wittgenstein&rsquos (re-)introduction of an extra-mathematical application criterion, which is used to distinguish mere &ldquosign-games&rdquo from mathematical language-games. &ldquo[I]t is essential to mathematics that its signs are also employed in mufti&rdquo, Wittgenstein states, for
[i]t is the use outside mathematics, and so the meaning [Bedeutung] of the signs, that makes the sign-game into mathematics. (i.e., a mathematical &ldquolanguage-game&rdquo RFM V, §2, 1942 LFM 140&ndash141, 169&ndash70)
As Wittgenstein says at (RFM V, §41, 1943),
[c]oncepts which occur in &lsquonecessary&rsquo propositions must also occur and have a meaning [Bedeutung] in non-necessary ones. (italics added)
If two proofs prove the same proposition, says Wittgenstein, this means that &ldquoboth demonstrate it as a suitable instrument for the same purpose&rdquo, which &ldquois an allusion to something outside mathematics&rdquo (RFM VII, §10, 1941 italics added).
As we have seen, this criterion was present in the Tractatus (6.211), but noticeably absent in the middle period. The reason for this absence is probably that the intermediate Wittgenstein wanted to stress that in mathematics everything is syntax and nothing is meaning. Hence, in his criticisms of Hilbert&rsquos &lsquocontentual&rsquo mathematics (Hilbert 1925) and Brouwer&rsquos reliance upon intuition to determine the meaningful content of (especially undecidable) mathematical propositions, Wittgenstein couched his finitistic constructivism in strong formalism, emphasizing that a mathematical calculus does not need an extra-mathematical application (PR §109 WVC 105).
There seem to be two reasons why the later Wittgenstein reintroduces extra-mathematical application as a necessary condition of a mathematical language-game. First, the later Wittgenstein has an even greater interest in the use of natural and formal languages in diverse &ldquoforms of life&rdquo (PI §23), which prompts him to emphasize that, in many cases, a mathematical &lsquoproposition&rsquo functions as if it were an empirical proposition &ldquohardened into a rule&rdquo (RFM VI, §23) and that mathematics plays diverse applied roles in many forms of human activity (e.g., science, technology, predictions). Second, the extra-mathematical application criterion relieves the tension between Wittgenstein&rsquos intermediate critique of set theory and his strong formalism according to which &ldquoone calculus is as good as another&rdquo (PG 334). By demarcating mathematical language-games from non-mathematical sign-games, Wittgenstein can now claim that, &ldquofor the time being&rdquo, set theory is merely a formal sign-game.
These considerations may lead us to say that (2^
That is to say: we can make the considerations lead us to that.
Or: we can say this and give this as our reason.
But if we do say it&mdashwhat are we to do next? In what practice is this proposition anchored? It is for the time being a piece of mathematical architecture which hangs in the air, and looks as if it were, let us say, an architrave, but not supported by anything and supporting nothing. (RFM II, §35)
It is not that Wittgenstein&rsquos later criticisms of set theory change, it is, rather, that once we see that set theory has no extra-mathematical application, we will focus on its calculations, proofs, and prose and &ldquosubject the interest of the calculations to a test&rdquo (RFM II, §62). By means of Wittgenstein&rsquos &ldquoimmensely important&rdquo &lsquoinvestigation&rsquo (LFM 103), we will find, Wittgenstein expects, that set theory is uninteresting (e.g., that the non-enumerability of &ldquothe reals&rdquo is uninteresting and useless) and that our entire interest in it lies in the &lsquocharm&rsquo of the mistaken prose interpretation of its proofs (LFM 16). More importantly, though there is &ldquoa solid core to all [its] glistening concept-formations&rdquo (RFM V, §16), once we see it as &ldquoas a mistake of ideas&rdquo, we will see that propositions such as &ldquo(2^
It must be emphasized, however, that the later Wittgenstein still maintains that the operations within a mathematical calculus are purely formal, syntactical operations governed by rules of syntax (i.e., the solid core of formalism).
It is of course clear that the mathematician, in so far as he really is &lsquoplaying a game&rsquo&hellip[is] acting in accordance with certain rules. (RFM V, §1)
To say mathematics is a game is supposed to mean: in proving, we need never appeal to the meaning [Bedeutung] of the signs, that is to their extra-mathematical application. (RFM V, §4)
Where, during the middle period, Wittgenstein speaks of &ldquoarithmetic [as] a kind of geometry&rdquo at (PR §109 & §111), the later Wittgenstein similarly speaks of &ldquothe geometry of proofs&rdquo (RFM I, App. III, §14), the &ldquogeometrical cogency&rdquo of proofs (RFM III, §43), and a &ldquogeometrical application&rdquo according to which the &ldquotransformation of signs&rdquo in accordance with &ldquotransformation-rules&rdquo (RFM VI, §2, 1941) shows that &ldquowhen mathematics is divested of all content, it would remain that certain signs can be constructed from others according to certain rules&rdquo (RFM III, §38). Hence, the question whether a concatenation of signs is a proposition of a given mathematical calculus (i.e., a calculus with an extra-mathematical application) is still an internal, syntactical question, which we can answer with knowledge of the proofs and decision procedures of the calculus.
3.6 Wittgenstein on Gödel and Undecidable Mathematical Propositions
RFM is perhaps most (in)famous for Wittgenstein&rsquos (RFM App. III) treatment of &ldquotrue but unprovable&rdquo mathematical propositions. Early reviewers said that &ldquo[t]he arguments are wild&rdquo (Kreisel 1958: 153), that the passages &ldquoon Gödel&rsquos theorem&hellip are of poor quality or contain definite errors&rdquo (Dummett 1959: 324), and that (RFM App. III) &ldquothrows no light on Gödel&rsquos work&rdquo (Goodstein 1957: 551). &ldquoWittgenstein seems to want to legislate [[q]uestions about completeness] out of existence&rdquo, Anderson said, (1958: 486&ndash87) when, in fact, he certainly cannot dispose of Gödel&rsquos demonstrations &ldquoby confusing truth with provability&rdquo. Additionally, Bernays, Anderson (1958: 486), and Kreisel (1958: 153&ndash54) claimed that Wittgenstein failed to appreciate &ldquoGödel&rsquos quite explicit premiss of the consistency of the considered formal system&rdquo (Bernays 1959: 15), thereby failing to appreciate the conditional nature of Gödel&rsquos First Incompleteness Theorem. On the reading of these four early expert reviewers, Wittgenstein failed to understand Gödel&rsquos Theorem because he failed to understand the mechanics of Gödel&rsquos proof and he erroneously thought he could refute or undermine Gödel&rsquos proof simply by identifying &ldquotrue in PM&rdquo (i.e., Principia Mathematica) with &ldquoproved/provable in PM&rdquo.
Interestingly, we now have two pieces of evidence (Kreisel 1998: 119 Rodych 2003: 282, 307) that Wittgenstein wrote (RFM App. III) in 1937&ndash38 after reading only the informal, &lsquocasual&rsquo (MS 126, 126&ndash127 Dec. 13, 1942) introduction of (Gödel 1931) and that, therefore, his use of a self-referential proposition as the &ldquotrue but unprovable proposition&rdquo may be based on Gödel&rsquos introductory, informal statements, namely that &ldquothe undecidable proposition [(R(q)q)] states&hellip that [(R(q)q)] is not provable&rdquo (1931: 598) and that &ldquo[(R(q)q)] says about itself that it is not provable&rdquo (1931: 599). Perplexingly, only two of the four famous reviewers even mentioned Wittgenstein&rsquos (RFM VII, §§19, 21&ndash22, 1941)) explicit remarks on &lsquoGödel&rsquos&rsquo First Incompleteness Theorem (Bernays 1959: 2 Anderson 1958: 487), which, though flawed, capture the number-theoretic nature of the Gödelian proposition and the functioning of Gödel-numbering, probably because Wittgenstein had by then read or skimmed the body of Gödel&rsquos 1931 paper.
The first thing to note, therefore, about (RFM App. III) is that Wittgenstein mistakenly thinks&mdashagain, perhaps because Wittgenstein had read only Gödel&rsquos Introduction&mdash(a) that Gödel proves that there are true but unprovable propositions of PM (when, in fact, Gödel syntactically proves that if PM is (omega)-consistent, the Gödelian proposition is undecidable in PM) and (b) that Gödel&rsquos proof uses a self-referential proposition to semantically show that there are true but unprovable propositions of PM.
For this reason, Wittgenstein has two main aims in (RFM App. III): (1) to refute or undermine, on its own terms, the alleged Gödel proof of true but unprovable propositions of PM, and (2) to show that, on his own terms, where &ldquotrue in calculus (Gamma)&rdquo is identified with &ldquoproved in calculus (Gamma)&rdquo, the very idea of a true but unprovable proposition of calculus (Gamma) is meaningless.
Thus, at (RFM App. III, §8) (hereafter simply &lsquo§8&rsquo), Wittgenstein begins his presentation of what he takes to be Gödel&rsquos proof by having someone say:
I have constructed a proposition (I will use &lsquoP&rsquo to designate it) in Russell&rsquos symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: &lsquoP is not provable in Russell&rsquos system&rsquo.
That is, Wittgenstein&rsquos Gödelian constructs a proposition that is semantically self-referential and which specifically says of itself that it is not provable in PM. With this erroneous, self-referential proposition P [used also at (§10), (§11), (§17), (§18)], Wittgenstein presents a proof-sketch very similar to Gödel&rsquos own informal semantic proof &lsquosketch&rsquo in the Introduction of his famous paper (1931: 598).
Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable. (§8)
The reasoning here is a double reductio. Assume (a) that P must either be true or false in Russell&rsquos system, and (b) that P must either be provable or unprovable in Russell&rsquos system. If (a), P must be true, for if we suppose that P is false, since P says of itself that it is unprovable, &ldquoit is true that it is provable&rdquo, and if it is provable, it must be true (which is a contradiction), and hence, given what P means or says, it is true that P is unprovable (which is a contradiction). Second, if (b), P must be unprovable, for if P &ldquois proved, then it is proved that it is not provable&rdquo, which is a contradiction (i.e., P is provable and not provable in PM). It follows that P &ldquocan only be true, but unprovable&rdquo.
To refute or undermine this &lsquoproof&rsquo, Wittgenstein says that if you have proved ( eg P), you have proved that P is provable (i.e., since you have proved that it is not the case that P is not provable in Russell&rsquos system), and &ldquoyou will now presumably give up the interpretation that it is unprovable&rdquo (i.e., &lsquoP is not provable in Russell&rsquos system&rsquo), since the contradiction is only proved if we use or retain this self-referential interpretation (§8). On the other hand, Wittgenstein argues (§8), &lsquo[i]f you assume that the proposition is provable in Russell&rsquos system, that means it is true in the Russell sense, and the interpretation &ldquoP is not provable&rdquo again has to be given up&rsquo, because, once again, it is only the self-referential interpretation that engenders a contradiction. Thus, Wittgenstein&rsquos &lsquorefutation&rsquo of &ldquoGödel&rsquos proof&rdquo consists in showing that no contradiction arises if we do not interpret &lsquoP&rsquo as &lsquoP is not provable in Russell&rsquos system&rsquo&mdashindeed, without this interpretation, a proof of P does not yield a proof of ( eg P) and a proof of ( eg P) does not yield a proof of P. In other words, the mistake in the proof is the mistaken assumption that a mathematical proposition &lsquoP&rsquo &ldquocan be so interpreted that it says: &lsquoP is not provable in Russell&rsquos system&rsquo&rdquo. As Wittgenstein says at (§11), &ldquo[t]hat is what comes of making up such sentences&rdquo.
This &lsquorefutation&rsquo of &ldquoGödel&rsquos proof&rdquo is perfectly consistent with Wittgenstein&rsquos syntactical conception of mathematics (i.e., wherein mathematical propositions have no meaning and hence cannot have the &lsquorequisite&rsquo self-referential meaning) and with what he says before and after (§8), where his main aim is to show (2) that, on his own terms, since &ldquotrue in calculus (Gamma)&rdquo is identical with &ldquoproved in calculus (Gamma)&rdquo, the very idea of a true but unprovable proposition of calculus (Gamma) is a contradiction-in-terms.
To show (2), Wittgenstein begins by asking (§5), what he takes to be, the central question, namely, &ldquoAre there true propositions in Russell&rsquos system, which cannot be proved in his system?&rdquo. To address this question, he asks &ldquoWhat is called a true proposition in Russell&rsquos system&hellip?&rdquo, which he succinctly answers (§6): &ldquo&lsquop&rsquo is true = p&rdquo. Wittgenstein then clarifies this answer by reformulating the second question of (§5) as &ldquoUnder what circumstances is a proposition asserted in Russell&rsquos game [i.e., system]?&rdquo, which he then answers by saying: &ldquothe answer is: at the end of one of his proofs, or as a &lsquofundamental law&rsquo (Pp.)&rdquo (§6). This, in a nutshell, is Wittgenstein&rsquos conception of &ldquomathematical truth&rdquo: a true proposition of PM is an axiom or a proved proposition, which means that &ldquotrue in PM&rdquo is identical with, and therefore can be supplanted by, &ldquoproved in PM&rdquo.
Having explicated, to his satisfaction at least, the only real, non-illusory notion of &ldquotrue in PM&rdquo, Wittgenstein answers the (§8) question &ldquoMust I not say that this proposition&hellip is true, and&hellip unprovable?&rdquo negatively by (re)stating his own (§§5&ndash6) conception of &ldquotrue in PM&rdquo as &ldquoproved/provable in PM&rdquo:
&lsquoTrue in Russell&rsquos system&rsquo means, as was said: proved in Russell&rsquos system and &lsquofalse in Russell&rsquos system&rsquo means: the opposite has been proved in Russell&rsquos system.
This answer is given in a slightly different way at (§7) where Wittgenstein asks &ldquomay there not be true propositions which are written in this [Russell&rsquos] symbolism, but are not provable in Russell&rsquos system?&rdquo, and then answers &ldquo&lsquoTrue propositions&rsquo, hence propositions which are true in another system, i.e. can rightly be asserted in another game&rdquo. In light of what he says in (§§5, 6, and 8), Wittgenstein&rsquos (§7) point is that if a proposition is &lsquowritten&rsquo in &ldquoRussell&rsquos symbolism&rdquo and it is true, it must be proved/provable in another system, since that is what &ldquomathematical truth&rdquo is. Analogously (§8), &ldquoif the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell&rsquos sense&rdquo, for &ldquo[w]hat is called &lsquolosing&rsquo in chess may constitute winning in another game&rdquo. This textual evidence certainly suggests, as Anderson almost said, that Wittgenstein rejects a true but unprovable mathematical proposition as a contradiction-in-terms on the grounds that &ldquotrue in calculus (Gamma)&rdquo means nothing more (and nothing less) than &ldquoproved in calculus (Gamma)&rdquo.
On this (natural) interpretation of (RFM App. III), the early reviewers&rsquo conclusion that Wittgenstein fails to understand the mechanics of Gödel&rsquos argument seems reasonable. First, Wittgenstein erroneously thinks that Gödel&rsquos proof is essentially semantical and that it uses and requires a self-referential proposition. Second, Wittgenstein says (§14) that &ldquo[a] contradiction is unusable&rdquo for &ldquoa prediction&rdquo that &ldquothat such-and-such construction is impossible&rdquo (i.e., that P is unprovable in PM), which, superficially at least, seems to indicate that Wittgenstein fails to appreciate the &ldquoconsistency assumption&rdquo of Gödel&rsquos proof (Kreisel, Bernays, Anderson).
If, in fact, Wittgenstein did not read and/or failed to understand Gödel&rsquos proof through at least 1941, how would he have responded if and when he understood it as (at least) a proof of the undecidability of P in PM on the assumption of PM&rsquos consistency? Given his syntactical conception of mathematics, even with the extra-mathematical application criterion, he would simply say that P, qua expression syntactically independent of PM, is not a proposition of PM, and if it is syntactically independent of all existent mathematical language-games, it is not a mathematical proposition. Moreover, there seem to be no compelling non-semantical reasons&mdasheither intra-systemic or extra-mathematical&mdashfor Wittgenstein to accommodate P by including it in PM or by adopting a non-syntactical conception of mathematical truth (such as Tarski-truth (Steiner 2000)). Indeed, Wittgenstein questions the intra-systemic and extra-mathematical usability of P in various discussions of Gödel in the Nachlass and, at (§19), he emphatically says that one cannot &ldquomake the truth of the assertion [&lsquoP&rsquo or &lsquoTherefore P&rsquo] plausible to me, since you can make no use of it except to do these bits of legerdemain&rdquo.
After the initial, scathing reviews of RFM, very little attention was paid to Wittgenstein&rsquos (RFM App. III and RFM VII, §§21&ndash22) discussions of Gödel&rsquos First Incompleteness Theorem (Klenk 1976: 13) until Shanker&rsquos sympathetic (1988b). In the last 22 years, however, commentators and critics have offered various interpretations of Wittgenstein&rsquos remarks on Gödel, some being largely sympathetic (Floyd 1995, 2001) and others offering a more mixed appraisal (Rodych 1999a, 2002, 2003 Steiner 2001 Priest 2004 Berto 2009a). Recently, and perhaps most interestingly, Floyd & Putnam (2000) and Steiner (2001) have evoked new and interesting discussions of Wittgenstein&rsquos ruminations on undecidability, mathematical truth, and Gödel&rsquos First Incompleteness Theorem (Rodych 2003, 2006 Bays 2004 Sayward 2005 and Floyd & Putnam 2006).
Preliminary mathematics and statistics
This course is an integral part for all students taking any of the MSc programmes in MSc Economics, MSc Development Economics, MSc International Finance and Development, MSc Economics and Environment, and MSc Political Economy of Development.
This course revises material usually taught in an undergraduate degree. A knowledge of algebra, calculus, and probability distributions is assumed. It is taught intensively for three weeks in September and consists of three parts: mathematics, statistics and computing. Students take an examination at the end of the course.
Topics covered include:
- Differentiation and optimisation techniques
- Simple differential equations
- Matrix algebra
- Statistical distributions
- Properties of estimators
- Interval estimation and hypothesis testing.
The other courses in the MSc build on this foundation.
This module will be delivered entirely online for 2020/21 via BLE. Students enrolled onto this module will be able to join the live Blackboard Collaborate session on BLE.
The first session is scheduled for Monday 7th September 2020. Please check the information sheet and timetable for the full schedule.