MAGIC Set theory. lecture 2
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1 MAGIC Set theory lecture 2 David Asperó University of East Anglia 22 October 2014
2 Recall from last time: Syntactical vs. semantical logical consequence Given a set T of formulas and a formula ', we write T = ' if and only if for every L structure M =(M, R) and every assignment ~a : Var! M, IF M = [~a] for every 2 T, THEN M = '[~a]. The relation = aims at capturing the notion of logical consequence : ' follows logically from T if and only if ' is true in every world in which T is true. = is often called the relation of logical consequence. First order in first order logic refers to the fact that variables range in the above definition only over the individuals of the universe of the relevant L structures M. In second order logic we can have variables that range over (arbitrary) subsets of the universe of the relevant L structures M. Etc.
3 Syntactical deduction Let T will be a set of formulas. We will view T as a set of axioms and deduce theorems from T : A theorem of T will be the final member n of a deduction =( 0, 1,..., n) from T, where we say that =( 0, 1,..., n) is a deduction from T if it is a finite sequence of L formulas and for every i, i is either in T, or i is a logical axiom of first order logic, or i is obtained form j and k, for some j, k < i, by the rule of Modus Ponens If '! and ', then (for all L formulas ', ). In other word, in this last case, there are j, k < i and an L formula ' such that j = ' and k is '! i.
4 Here, a logical axiom is a member of a certain infinite easily specifiable list of formulas that express logical / completely general truths. Typical members of this list are for example, '! (! ') for all formulas ',, or ' _ ' for all formulas '. Indeed, we see it as a general truth that if ' is true, then it is true that if is true then ' is true. And we see it as a general truth that for every ' either ' is true or ' is true. 1 This list of axioms is not unique: Many different lists of axioms give rise to the same system of logic. 1 If we are classical logicians. There are weakening / versions of classical first order logic in which ' _ ', also known as Law of Excluded Middle, is not true for some choices of '.
5 If ' is a theorem from T, we write T ` ' ` is often called the relation of logical derivability.
6 A priori, = and ` look like quite different relations, aimed at capturing two apparently different notions: The notion of logical (semantical) consequence and the notion of deductibility in a reasonable calculus. However,
7 Theorem (Completeness theorem for first order logic) (K. Gödel, 1930 s) ==`
8 A theory T is consistent if no contradiction (say, 9x (x = x)) can be derived from it: T 0 9x (x = x) A theory is inconsistent (i.e., not consistent) if and only if it is trivial, in the sense that it proves everything. By the completeness theorem the following are equivalent: T is consistent. There is an L structure M such that M = T (T is true in some world).
9 We ll be interested in whether or not T ` and a given sentence. for a given theory T The following are equivalent again by (the contrapositive of) the completeness theorem: T 0 There is an L structure M such that M = T but M =.
10 Axiomatic set theory: ZFC Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for the Axiom of Choice. The objects of set theory are sets. As in any axiomatic theory, they are not defined (they are feature less objects; in the context of the theory there is nothing to them apart from what the theory says). ZFC expresses facts about sets expressible in the first order language of set theory. The same is true for any other first order theory in the language of set theory, like ZF, ZFC+ There is a supercompact cardinal + ZFC+GCH, ZFC+V = L, ZFC+PFA,...
11 Axiomatic set theory: ZFC Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for the Axiom of Choice. The objects of set theory are sets. As in any axiomatic theory, they are not defined (they are feature less objects; in the context of the theory there is nothing to them apart from what the theory says). ZFC expresses facts about sets expressible in the first order language of set theory. The same is true for any other first order theory in the language of set theory, like ZF, ZFC+ There is a supercompact cardinal + ZFC+GCH, ZFC+V = L, ZFC+PFA,...
12 Axiomatic set theory: ZFC Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for the Axiom of Choice. The objects of set theory are sets. As in any axiomatic theory, they are not defined (they are feature less objects; in the context of the theory there is nothing to them apart from what the theory says). ZFC expresses facts about sets expressible in the first order language of set theory. The same is true for any other first order theory in the language of set theory, like ZF, ZFC+ There is a supercompact cardinal + ZFC+GCH, ZFC+V = L, ZFC+PFA,...
13 Most ZFC axioms will be axioms saying that certain classes (built out of given sets) are actual sets (they are objects in the set theoretic universe): Axiom 0, The Axiom of unordered pairs, Union set Axiom, Power set Axiom, Axiom Scheme of Separation, Axiom Scheme of Replacement and Axiom of Infinity will be of this kind. Here, a class is any collection of objects, where this collection is definable possibly with parameters. For example the class of all sets. A proper class will be a class which is not a set. ZFC will also have an axiom guaranteeing the existence of sets with a given property, even if these sets are not definable: The Axiom of Choice We will also have two structural axioms: Axiom of Extensionality and Axiom of Foundation.
14 Most ZFC axioms will be axioms saying that certain classes (built out of given sets) are actual sets (they are objects in the set theoretic universe): Axiom 0, The Axiom of unordered pairs, Union set Axiom, Power set Axiom, Axiom Scheme of Separation, Axiom Scheme of Replacement and Axiom of Infinity will be of this kind. Here, a class is any collection of objects, where this collection is definable possibly with parameters. For example the class of all sets. A proper class will be a class which is not a set. ZFC will also have an axiom guaranteeing the existence of sets with a given property, even if these sets are not definable: The Axiom of Choice We will also have two structural axioms: Axiom of Extensionality and Axiom of Foundation.
15 A classification of the ZFC axioms 1 Structural axioms: Axioms of Extensionality, Axiom of Foundation. 2 Constructive set existence axioms: Axiom 0, The Axiom of unordered pairs, Union set Axiom, Power set Axiom, Axiom Scheme of Separation, Axiom Scheme of Replacement and Axiom of Infinity. 3 Non constructive set existence axiom: Axiom of Choice.
16 The axioms Axiom of Extensionality: Two sets are equal if and only if they have the same elements: 8x8y(x = y $8z(z 2 x $ z 2 y)) In other words: the identity of a set is completely determined by its members: The sets ; {(a, b, c, n) :a n +b n = c n, a, b, c, n 2 N, a, b, c 2, n 3} are the same set.
17 Axiom 0: ; exists. 9x8y(y 2 x $ y 6= y) (of course y 6= y abbreviates (y = y)). Strictly speaking this axiom is not needed: It follows from the other axioms. It is convenient to postulate it at this point, though.
18 In the theory given by the Axiom of Extensionality together with Axiom 0 we can only prove the existence of one set: Not so interesting yet. The theory T = { Axiom 0, Axiom of Extensionality } surely is consistent: For any set a, But ({a, b}, ;) 6 = T if a 6= b. ; ({a}, ;) = T
19 Axiom of unordered pairs: For any sets x, y there is a set whose members are exactly x and y; in other words, {x, y} exists. 8x8y9z8w(w 2 z $ (w = x _ w = y)) Of course: If x = y, then {x, y} = {x}. [Prove this using the Axiom of Extensionality.] Recall: Definition: (x, y) ={{x}, {x, y}}
20 The theory laid down so far gives us already the existence of infinitely many sets! : ;, {;}, {{;}}, {{{;}}}, {{{{;}}}}, {;, {;}}, {;, {;, {;}}}, {{;}, {;, {;}}}, {;, {;, {;, {;}}}},... With the definition of the natural numbers given in lecture 1, these sets are: 0, 1, {1} =(0, 0), {{1}} = {(0, 0)}, ((0, 0), (0, 0)), 2=(0, 1), {0, 2}, {1, 2} =(0, 1), {0, {0, 2}},... All sets whose existence is proved by the theory given so far have at most two elements (!). This theory proves the existence of (a, b) for all a, b.
21 Union set Axiom: For every set x, [ x = {y :(9w)(w 2 x ^ y 2 w)} exists. 8x9v8y(y 2 v $ (9w)(w 2 x ^ y 2 w)) S x is the set consisting of all the members of members of x, SS x is the set of all the members of members of members of x, etc. Notation: Given sets x, y, x [ y = {a : a 2 x _ a 2 y} = S {x, y}. Note: Given sets x, y, x [ y exists (by the Axiom of unordered pairs and the Union set Axiom).
22 With the theory given so far we can prove the existence of: {0}[{1, 2} = {0, 1, 2} = 3, {0, 1, 2}[{3} = {0, 1, 2, 3} = 4, {0, 1, 2, 3}[{4} = {0, 1, 2, 3, 4} = 5,... So we can prove the existence of every individual natural number! Similarly, we can prove the existence of every finite set of natural numbers, every ordered pair of natural numbers, every tuple of natural numbers, every finite set of tuples of natural numbers,... However: All particular sets proved to exist by the theory given so far are finite.
23 Notation: z x means: of x. Every member of z is a member Power set Axiom: For every x there is y whose elements are exactly those z which are a subset of x: 8x9y8z(z 2 y $ (8w)(w 2 z! w 2 x)) Notation: For every a, P(a) ={z : z a}. The Power set Axiom says that P(a) is a set whenever a is a set.
24 With the theory T laid down so far we can prove the existence of P(n) for any particular n 2 N. For example: P(0) ={;} = 1 P(1) ={;, {;}} = 2 P(2) ={;, {;}, {{;}}, {;, {;}}} 6= 4...
25 T is consistent: 2 Let (X n ) n2n defined recursively by X 0 = {;} X n+1 = X n [ {{a, b} : a, b 2 X n }[{ S a : a 2 X n } [ {P(a) :a 2 X n } Then ( S n2n X n, 2) = T. Actually it would be enough to start with ; and take X n+1 = P(X n ) at each stage n + 1. Note: All particular sets proved to exist by T are still finite. 2 Isn t it?
26 T is consistent: 2 Let (X n ) n2n defined recursively by X 0 = {;} X n+1 = X n [ {{a, b} : a, b 2 X n }[{ S a : a 2 X n } [ {P(a) :a 2 X n } Then ( S n2n X n, 2) = T. Actually it would be enough to start with ; and take X n+1 = P(X n ) at each stage n + 1. Note: All particular sets proved to exist by T are still finite. 2 Isn t it?
27 Axiom Scheme of Separation: Given any set X and any first order property P, {y 2 X : P(y)} exists; in other words: any definable subclass of a set exists as a set. 8x8v 0,...v n 9y8z(z 2 y $ (y 2 x ^ '(y, x, v 0,...v n ))) for every L formula '(y, x, v 0,...,v n ) such that none of x, y, z occur as free variables in '.
28 In the theory laid down so far we can prove the existence, for all x, y, of x y = {(a, b) :a 2 x, b 2 y}, and much more. x y exists: Let z = x [ y, which we know exists in our theory. Note that x y is a definable sub-collection of P(P(x [ y)). Hence x y exists using Power set twice and Separation once.
29 For a formula '(v 0,...v n, u, v), '(v 0,...v n, u, v) is functional is an abbreviation of the formula expressing for all u there is at most one v such that '(v 0,...v n, u, v) Axiom Scheme of Replacement: Given any set X and any definable (class) function F, range(f X) is a set: For all x, v 0,...,v n, if '(v 0,...v n, u, v) is functional, then there is y such that for all v, v 2 y if and only if there is some u 2 x such that '(v 0,...v n, u, v), for every formula '(v 0,...v n, u, v) such that none of x, y, z occur as free variables in '.
30 Caution: The Axiom schemes of Separation and Replacement are not axioms but infinite sets of axioms (!) However, it is obviously possible to write down a computer program which, given a sentence, recognises whether or not belongs to either of these schemes.
31 Given a set X such that a 6= ; for all a 2 X, achoice function for X is a function f with dom(f )=X and such that f (a) 2 a for all a 2 X. Axiom of Choice (AC): Every set consisting of nonempty sets has a choice function. Exercise: Write down a sentence expressing the Axiom of Choice. AC is needed in a lot of mathematics. For example, to prove that every vector space has a basis, that there are sets of reals which are not Lebesgue measurable, etc.
32 Nevertheless, historically AC has often been seen with suspicion: Finite sets clearly have choice functions, but if X is infinite, where did the choice function for X come from? Also: AC has strange consequences : It is possible to decompose a sphere S into finitely many pieces and rearrange them, without changing their volumes in fact by moving them around and rotating them, and without running into one another, in such a way that we obtain two spheres with the same volume as S! (Banach Tarski paradox) The pieces are not Lebesgue measurable. The Banach Tarski is not an actual paradox, in the sense that Russel s paradox is, but a counterintuitive fact.
33 Nevertheless, historically AC has often been seen with suspicion: Finite sets clearly have choice functions, but if X is infinite, where did the choice function for X come from? Also: AC has strange consequences : It is possible to decompose a sphere S into finitely many pieces and rearrange them, without changing their volumes in fact by moving them around and rotating them, and without running into one another, in such a way that we obtain two spheres with the same volume as S! (Banach Tarski paradox) The pieces are not Lebesgue measurable. The Banach Tarski is not an actual paradox, in the sense that Russel s paradox is, but a counterintuitive fact.
34 AC has interesting equivalent formulations (modulo the rest of ZF). For example AC is equivalent to For every two nonempty sets A, B, either A apple B or B < A.
35 The Axiom of Foundation: If X 6= ; is a set, there is some a 2 X such that b /2 a for every b 2 X. In other word: Every nonempty sets has some 2 minimal element. Modulo the other axioms (in particular AC), the following are equivalent: Foundation There are no x 0, x 1,...,x n, x n+1,...such that...2 x n+1 2 x n x 1 2 x 0.
36 Idea behind Foundation: Sets are generated at different stages. If a set X is generated at stage, then all members of X have been generated at some stage before. Foundation, together with Extensionality, of course, is perhaps the most fundamental axiom in set theory (!).
37 As with AC, one could perhaps also complain: Where did the 2 minimal element a of X come from? But wait. a was already in X. If you remove a from X, X is no longer X! In fact, most people like Foundation: It says that the universe is generated in an orderly fashion. And it provides a very convenient tool to use in proofs, which we will be using all the time: Induction.
38 As with AC, one could perhaps also complain: Where did the 2 minimal element a of X come from? But wait. a was already in X. If you remove a from X, X is no longer X! In fact, most people like Foundation: It says that the universe is generated in an orderly fashion. And it provides a very convenient tool to use in proofs, which we will be using all the time: Induction.
39 As with AC, one could perhaps also complain: Where did the 2 minimal element a of X come from? But wait. a was already in X. If you remove a from X, X is no longer X! In fact, most people like Foundation: It says that the universe is generated in an orderly fashion. And it provides a very convenient tool to use in proofs, which we will be using all the time: Induction.
40 Let (V n ) n2n be defined by recursion as follows. V 0 = ; V n+1 = P(V n ) The theory laid down so far, T = Ax0+ Extensionality + Unordered Pairs + Union + Power Set + Separation + Replacement + AC + Foundation, is consistent. 3 In fact ( [ n V n, 2) = T Still, all sets proved by T to exist are finite. In fact, ( [ n V n, 2) = Every set is finite 3 Isn t it?
41 Finite? For the moment let us say: X is finite if and only if for every a 2 X, X \{a} < X. X is infinite if and only if X is not finite. The above is not the official definition of finite but is equivalent to the official definition. But it makes things easier to deal with the above definition (which does not involve the notion of ordinal, which we haven t defined yet).
42 Axiom of Infinity: There is an infinite set. Definition: Given a set x, S(x) =x [{x} (the successor of x). So, S(0) =1, S(1) =2,... S(n) =n + 1. The Axiom of Infinity is equivalent to: (9x)(; 2x ^ (8y)(y 2 x! S(y) 2 x)) This is also phrased as: There is an inductive set. Note: Every inductive set is infinite (in our present sense). Proof: Try to prove it yourself without help.
43 Axiom of Infinity: There is an infinite set. Definition: Given a set x, S(x) =x [{x} (the successor of x). So, S(0) =1, S(1) =2,... S(n) =n + 1. The Axiom of Infinity is equivalent to: (9x)(; 2x ^ (8y)(y 2 x! S(y) 2 x)) This is also phrased as: There is an inductive set. Note: Every inductive set is infinite (in our present sense). Proof: Try to prove it yourself without help.
44 One could also define is an ordinal (which we ll do soon). Then: Definition: A natural number is an ordinal such that 1 is either ; or of the form S(y) for some y 2 and 2 for every x 2, x is either ; or of the form S(y) for some y 2. The Axiom of Infinity is then equivalent to: Axiom of Infinity : The class of all natural numbers is a set. In other words: There is some x such that for all y, y 2 x if and only y is a natural number. Note that Axiom of Infinity is a constructive set existence axiom, whereas Axiom of Infinity was not, strictly speaking. Axiom of Infinity and Axiom of Infinity are equivalent modulo the other axioms.
45 The Axiom of Infinity completes the list of ZFC axioms. Notice the big leap when adding Infinity to the list of axioms. ZFC certainly proves the existence of infinite sets, by design! Before adding Infinity we had a theory T which surely was consistent (since ( S n2n V n, 2) = T ). Now, with the addition of Infinity, it s not so obvious that ZFC is consistent.... Challenge: Construct a model of ZFC.
46 ZFC vs PA PA (Peano Arithmetic): A first order theory for (N, S, +,, 0), where S(n) = n + 1 (in the language of arithmetic, i.e., the language with S, +,, 0): 8x(S(x) 6= 0) 8x, y, (S(x) =S(y) $ x = y) 8x(x + 0 = x) 8x, y(x + S(y) =S(x + y)) 8x(x 0 = 0) 8x, y(x S(y) =x y + x) 8ȳ(('(0, ȳ) ^ (8x('(x, ȳ)! '(S(x), ȳ)))!8x'(x, ȳ)) for every first order formula '(x, ȳ) in the language of arithmetic (First order Induction Axiom Scheme)
47 First order arithmetical facts can be expressed in this language: is distributive with respect to +, Fermat s last theorem, Goldbach s conjecture,... PA does prove many facts about (N, S, +,, 0). But it does not prove everything! Theorem (Gödel, 1930 s, Incompleteness Theorem (special case)) If PA is consistent then there is a sentence in the language of arithmetic such that PA 0 and PA 0
48 Gödel s Incompleteness theorem(s), in their general formulation, are very profound facts that we will look back into in a moment.
49 The sentence in the Incompleteness Theorem does not express any fact that mathematicians would have looked into prior to proving the incompleteness theorem. is designed for the purpose of the proof only.
50 Notation: Given a set X and n 2 N, let [X] n = {a X : a = n } Consider the following statement HP: For all n, k, m there is some N such that for every colouring f :[N] n into k colours there is some Y N such that Y has at least m many members and at least min(y ) many members and such that all members of [Y ] n have the same colour under f. Here, n, k, m and N range over natural numbers. HP can be easily expressed by a sentence, which I will call HP, in the language of arithmetic. ZFC proves that (N, S, +,, 0) = HP.
51 Notation: Given a set X and n 2 N, let [X] n = {a X : a = n } Consider the following statement HP: For all n, k, m there is some N such that for every colouring f :[N] n into k colours there is some Y N such that Y has at least m many members and at least min(y ) many members and such that all members of [Y ] n have the same colour under f. Here, n, k, m and N range over natural numbers. HP can be easily expressed by a sentence, which I will call HP, in the language of arithmetic. ZFC proves that (N, S, +,, 0) = HP.
52 On the other hand: Theorem (L. Harrington and J. Paris, 1977): If PA is consistent, then PA 0 HP
53 ZFC vs PA Consider the theory T =(ZFC \{Infinity}) [ { Infinity}. It turns out that T and PA are essentially the same theory: There are effective translation procedures '! (') between the sentences in the language of set theory and the sentences in the language of arithmetic and! ( ) between the sentences in the language of arithmetic and the sentences in the language of set theory such that for all ',, T ` ' if and only if PA ` (') PA ` if and only if T ` ( )
54 The Harrington Paris theorem gives an example of a simple natural (purely combinatorial) statement talking only about finite sets which is true if there is an infinite set but need not be true if there are no infinite sets (!) Other examples have been found since then.
55 The Harrington Paris theorem gives an example of a simple natural (purely combinatorial) statement talking only about finite sets which is true if there is an infinite set but need not be true if there are no infinite sets (!) Other examples have been found since then.
56 The consistency question We pointed out that T = ZFC \{Infinity} was surely consistent, based on the fact that ( S n2n V n, 2) = T (assuming, in our metatheory, that P(a) exists for every a, that N exists, that the recursive construction of F =(V n ) n2n is well defined class function, and that S range(f) exists, i.e., assuming something like ZFC in our metatheory!) Question: Can we prove, in T (equivalently, in PA), that T is consistent? Can we prove, in ZFC, that ZFC is consistent?
57 The consistency question We pointed out that T = ZFC \{Infinity} was surely consistent, based on the fact that ( S n2n V n, 2) = T (assuming, in our metatheory, that P(a) exists for every a, that N exists, that the recursive construction of F =(V n ) n2n is well defined class function, and that S range(f) exists, i.e., assuming something like ZFC in our metatheory!) Question: Can we prove, in T (equivalently, in PA), that T is consistent? Can we prove, in ZFC, that ZFC is consistent?
58 The consistency question We pointed out that T = ZFC \{Infinity} was surely consistent, based on the fact that ( S n2n V n, 2) = T (assuming, in our metatheory, that P(a) exists for every a, that N exists, that the recursive construction of F =(V n ) n2n is well defined class function, and that S range(f) exists, i.e., assuming something like ZFC in our metatheory!) Question: Can we prove, in T (equivalently, in PA), that T is consistent? Can we prove, in ZFC, that ZFC is consistent?
59 The consistency question We pointed out that T = ZFC \{Infinity} was surely consistent, based on the fact that ( S n2n V n, 2) = T (assuming, in our metatheory, that P(a) exists for every a, that N exists, that the recursive construction of F =(V n ) n2n is well defined class function, and that S range(f) exists, i.e., assuming something like ZFC in our metatheory!) Question: Can we prove, in T (equivalently, in PA), that T is consistent? Can we prove, in ZFC, that ZFC is consistent?
60 The above questions do make sense: Both T and PA have enough expressive power to make T is consistent, PA is consistent, etc. expressible in the theory: For example, we can code formulas, proofs, and other syntactical notions as natural numbers and reduce a statement like PA is consistent to an arithmetical statement (some specific, but extremely complex, diophantine equation p( x) =0 does not have solutions). It then makes sense to ask whether T proves that p( x) =0 does not have solutions.
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