MAGIC Set Theory Lecture 2

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MAGIC Set theory

lecture 2

David Aspero´

University of East Anglia

18 October 2018 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).

Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for the Axiom of Choice.

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, ... 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. 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. 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. The axioms

Axiom of Extensionality: Two sets are equal if and only if they have the same elements:

x y(x = y z(z x z y)) 8 8 $8 2 $ 2 In other words, the identity of a set is completely determined by its members:

The sets

• ; (a, b, c, n):an+bn = cn, a, b, c, n N, a, b, c 2, n 3 • { 2 } are the same set. Axiom 0: exists. ; x y(y x y = y) 9 8 2 $ 6 (of course y = y abbreviates (y = y)). 6 ¬

Strictly speaking this axiom is not needed: It follows from the other axioms.

It is convenient to postulate it at this point, though. 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,

( a , ) = T { } ; | But ( a, b , ) = T if a = b. { } ; 6| 6 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.

x y z w(w z (w = x w = y)) 8 8 9 8 2 $ _ Of course: If x = y, then x, y = x . { } { } [Prove this using the Axiom of Extensionality.]

Recall: Deﬁnition: (x, y)= x , x, y {{ } { }} 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 (!). In fact this theory is consistent together with the sentence that says “Every set has at most two elements” (starting from and ; closing under unordered pair gives rise to a model of it where “Every set has at most two elements” also holds).

This theory proves the existence of (a, b) for all a, b. Union set Axiom: For every set x,

x = y :(w)(w x y w) { 9 2 ^ 2 } [ exists.

x v y(y v ( w)(w x y w)) 8 9 8 2 $ 9 2 ^ 2 x is the set consisting of all the members of members of x, x is the set of all the members of members of members of S x, etc. SS Notation: Given sets x, y, x y = a : a x a y = x, y . [ { 2 _ 2 } { } S Note: Given sets x, y, x y exists (by the Axiom of unordered [ pairs and the Union set Axiom). 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. Notation: z x means: Every member of z is a member ✓ of x.

Power set Axiom: For every x there is y whose elements are exactly those z which are a subset of x:

x y z(z y ( w)(w z w x)) 8 9 8 2 $ 8 2 ! 2 Notation: For every a, (a)= z : z a . P { ✓ } The Power set Axiom says that (a) is a set whenever a is a P set. With the theory T laid down so far we can prove the existence of (n) for any particular n N. P 2

For example: (0)= = 1 • P {;} (1)= , = 2 • P {; {;}} (2)= , , , , = 4 • P {; {;} {{;}} {; {;}}} 6 ... • 1 T is consistent: Let (Xn)n N deﬁned recursively by 2 X = • 0 {;} X = X a, b : a, b X a : a • n+1 n [{{ } 2 n}[{ 2 X (a):a X n}[{P 2 n} S Then ( n N Xn, ) = T . 2 2 | S Actually it would be enough to start with and take ; X = (X ) at each stage n + 1. n+1 P n

Note: All particular sets proved to exist by T are still ﬁnite.

1Isn’t it? 1 T is consistent: Let (Xn)n N deﬁned recursively by 2 X = • 0 {;} X = X a, b : a, b X a : a • n+1 n [{{ } 2 n}[{ 2 X (a):a X n}[{P 2 n} S Then ( n N Xn, ) = T . 2 2 | S Actually it would be enough to start with and take ; X = (X ) at each stage n + 1. n+1 P n

Note: All particular sets proved to exist by T are still ﬁnite.

1Isn’t it? Axiom Scheme of Separation: Given any set X and any ﬁrst order property P, y X : P(y) { 2 } exists; in other words: any deﬁnable subclass of a set exists as a set.

x v ,...v y z(z y (z x '(z, x, v ,...v ))) 8 8 0 n9 8 2 $ 2 ^ 0 n for every –formula '(y, x, v ,...,v ) such that none of x, y, z L 0 n occur as free variables in '. In the theory laid down so far we can prove the existence, for all x, y, of x y = (a, b):a x, b y , ⇥ { 2 2 } 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 ( (x y)). ⇥ P P [ Hence x y exists using Power set twice and Separation once. ⇥ For a formula '(v0,...vn, u, v),‘'(v0,...vn, u, v) is functional’ is an abbreviation of the formula expressing

“for all u there is at most one v such that '(v0,...vn, u, v)”

Axiom Scheme of Replacement: Given any set X and any deﬁnable (class)–function F, range(F X) is a set: [F X is the restriction of F to X, i.e. (a, b) F : a X ] { 2 2 }

“For all x, v0,...,vn, if '(v0,...vn, u, v) is functional, then there is y such that for all v, v y if and only if there is some u x 2 2 such that '(v0,...vn, u, v),” for every formula '(v0,...vn, u, v) such that none of x, y, z occur as free variables in '. 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. Given a set X such that a = for all a X,achoice function 6 ; 2 for X is a function f with dom(f )=X and such that f (a) a for 2 all a X. 2 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. 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. 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?

The Banach–Tarski is not an actual paradox, in the sense that Russel’s paradox is, but a counterintuitive fact. 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 B or B < A ”. | || | | | | | The Axiom of Foundation: If X = is a set, there is some 6 ; a X such that b / a for every b X. 2 2 2

In other word: Every nonempty sets has some –minimal element. 2

Modulo the other axioms (in particular AC), the following are equivalent: Foundation • There are no x , x ,...,x , x ,...such that • 0 1 n n+1 ... x x ... x x . 2 n+1 2 n 2 2 1 2 0 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 (!). As with AC, one could perhaps also complain: Where did the –minimal element a of X come from? But wait. a was already 2 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. As with AC, one could perhaps also complain: Where did the –minimal element a of X come from? But wait. a was already 2 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. Let (Vn)n N be deﬁned by recursion as follows. 2 V = • 0 ; V = (V ) • n+1 P n The theory laid down so far, T = Ax0+ Extensionality + Unordered Pairs + Union + Power Set + Separation + Replacement + AC + Foundation, is consistent.2 In fact

( V , ) = T n 2 | n [ Still, all sets proved by T to exist are ﬁnite. In fact,

( V , ) = “Every set is ﬁnite” n 2 | n [

2Isn’t it? Finite?

For the moment let us say:

X is finite if and only if for every a X, X a < X . 2 | \{ }| | | 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). 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:

( x)( x ( y)(y x S(y) x)) 9 ;2 ^ 8 2 ! 2 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. ⇤ 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:

( x)( x ( y)(y x S(y) x)) 9 ;2 ^ 8 2 ! 2 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. ⇤ One could also deﬁne “↵ is an ordinal” (which we’ll do soon). Then:

Deﬁnition:Anatural number is an ordinal ↵ such that 1 ↵ is either or of the form S(y) for some y ↵ and ; 2 2 for every x ↵, x is either or of the form S(y) for some 2 ; 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 x if and only y is a natural number. 2 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 turn out to be equivalent modulo the other axioms.

This, for one thing, shows that the previous classiﬁcation of ‘existence axioms’ into constructive and non-constructive existence axioms is perhaps not a very good one: The constructive aspect of an axiom may depend on the (remaining) background theory. 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 ( n N Vn, ) = T ). 2 2 | S Now, with the addition of Inﬁnity, it’s not so obvious that ZFC is consistent... .

Challenge: Construct a model of ZFC. 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): · x(S(x) = 0) • 8 6 x, y, (S(x)=S(y) x = y) • 8 $ x(x + 0 = x) • 8 x, y(x + S(y)=S(x + y)) • 8 x(x 0 = 0) • 8 · x, y(x S(y)=x y + x) • 8 · · y¯(('(0, y¯) ( x('(x, y¯) '(S(x), y¯))) x'(x, y¯)) • 8 ^ 8 ! !8 for every ﬁrst order formula '(x, y¯) in the language of arithmetic

(First order Induction Axiom Scheme) 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 (Godel,¨ 1930’s, Incompleteness Theorem (special case)) If PA is consistent then there is a sentence in the language of arithmetic such that

PA and • 0 PA 0 • ¬ Godel’s¨ Incompleteness theorem(s), in their general formulation, are very profound facts that we will look back into in a moment.

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. Notation: Given a set X and n N, let 2 [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 of [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. · | Notation: Given a set X and n N, let 2 [X]n = a X : a = n { ✓ | | | |} Consider the following statement HP:

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. · | On the other hand:

Theorem (L. Harrington and J. Paris, 1977): If PA is consistent, then PA 0 HP 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 ( ) • ` ` 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. 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. The consistency question

We pointed out that T = ZFC Inﬁnity was ‘surely’ consistent, \{ } based on the fact that ( n N Vn, ) = T 2 2 | S (assuming, in our metatheory, that (a) exists for every a, that P N exists, that the recursive construction of F =(Vn)n N is 2 well–deﬁned class–function, and that range(F) exists, i.e., assuming something like ZFC in our metatheory !) S

Question: Can we prove, in T , that T is consistent? Can we prove, in ZFC, that ZFC is consistent? The consistency question

Question: Can we prove, in T , that T is consistent? Can we prove, in ZFC, that ZFC is consistent? 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, polynomial equation p(x¯)=0 in many variables does not have solutions). It then makes sense to ask whether T proves that p(x¯)=0 does not have solutions.