MAGIC Set theory lecture 6 David Aspero´ University of East Anglia 15 November 2018 Recall: We defined (V : ↵ Ord) by recursion on Ord: ↵ 2 V = • 0 ; V = (V ) • ↵+1 P ↵ V = V : β<δ if δ is a limit ordinal. • δ { β } S (V : ↵ Ord) is called the cumulative hierarchy. ↵ 2 We saw: Proposition V↵ is transitive for every ordinal ↵. Proposition For all ↵<β, V V . ↵ ✓ β Definition For every x V , let rank(x) be the first ↵ such that 2 ↵ Ord ↵ x V . 2 2 ↵+1 S Definition For every set x, the transitive closure of x, denoted by TC(x), is X : n <! where { n } X = x S • 0 X = X • n+1 n So TC(x)=Sx x x x ... [ [ [ [ S SS SSS [Exercise: TC(x) is the –least transitive set y such that x y. ✓ ✓ In other words, TC(x)= y : y transitive, x y .] { ✓ } T Definition V denotes the class of all sets; that is, V = x : x = x . { } Definition WF = V : ↵ Ord : The class of all x such that x V for { ↵ 2 } 2 ↵ some ordinal ↵. S Note: WF is a transitive class: y x V implies y V since 2 2 ↵ 2 ↵ V↵ is transitive. Proposition If x WF, then there is some ↵ Ord such that x V . ✓ 2 ✓ ↵ Proof. Define the function F(y)=min γ y V . By the assumption { | 2 γ} on x, F is a well-defined function there. By Replacement γ y x : γ = F(y) is a set, and by Union it has a { |9 2 } supremum ↵. Therefore x V . ✓ ↵ Corollary If x WF, then x WF. ✓ 2 Definition V denotes the class of all sets; that is, V = x : x = x . { } Definition WF = V : ↵ Ord : The class of all x such that x V for { ↵ 2 } 2 ↵ some ordinal ↵. S Note: WF is a transitive class: y x V implies y V since 2 2 ↵ 2 ↵ V↵ is transitive. Proposition If x WF, then there is some ↵ Ord such that x V . ✓ 2 ✓ ↵ Proof. Define the function F(y)=min γ y V . By the assumption { | 2 γ} on x, F is a well-defined function there. By Replacement γ y x : γ = F(y) is a set, and by Union it has a { |9 2 } supremum ↵. Therefore x V . ✓ ↵ Corollary If x WF, then x WF. ✓ 2 Theorem (ZF) V = WF Proof: Let x be some set, and let y be TC(x). It is enough to show that y WF, since that implies x WF and therefore ✓ ✓ x WF. Let y = z y z / WF . If y is not empty, by 2 ⇤ { 2 | 2 } ⇤ Foundation it has some -minimal a. By transitivity of y, a y 2 ✓ and so if z a, z WF. But this means that a WF and 2 2 ✓ therefore a WF. 2 Therefore y ⇤ = ?, so y WF as wanted. ✓ The picture of the universe provided by V = WF is a very appealing and very natural one (once one has come across it, at least). This picture of the universe of all sets, and the fact that ZF implies V = WF, is the main source of intrinsic justifications of the ZF axioms. Inner models and relativization Let (M, M ) be a submodel, or inner model, defined by a 2 formula ⇥(x); in other words, M = a :⇥(a) and, for all a, { } b M, a M b if and only if a b (we usually leave out M and 2 2 2 2 write M instead of (M, M )). 2 (Examples: V , WF, L, HOD, ...). We define the relativization to M of a formula '(~x), to be denoted 'M (~x), in the following manner. (x y)M is x y. • 2 2 (x = y)M is x = y. • (' ' )M is 'M 'M . • 0 _ 1 0 _ 1 ( ')M is ('M ). • ¬ ¬ (( x)('(x¯))M is x(⇥(x) 'M (x¯)). We may also write • 8 8 ! ( x M)'M (x). 8 2 Note: Given a formula '(x ,...,x ) and a ,...,a M, 0 n 0 n 2 'M (a ,...,a ) holds if and only if M = '[a ,...,a ]. [Easy to 0 n | 0 n prove by induction on the complexity of '.] Notation: If (N, E) is a structure in the language of set theory, M is an inner model defined by a formula ⇥(x) possibly with parameters (i.e., M =(N, E (M M)), where ⇥ M = a N :(N, E) =⇥(a) ), and we want / need to { 2 | } emphasise that M is the inner model defined by ⇥(x) as defined within (N, E), then we often write MN instead of M. Example: WFM WF Note: For every ordinal ↵, V↵ = V↵ (here V↵ refers to the set, definable from the parameter ↵, with the definition that we have seen). (This note is pertinent under ZF Foundation , in which \{ } context V = WF might not be true.) Many facts about the universe V are inherited by reasonable submodels. For example: Lemma Suppose M is a transitive set or a transitive proper class. Then M = Axiom of Extensionality. | Proof: Let a, b M and suppose M =( x)(x a x b) 2 | 8 2 $ 2 (this of course is shorthand for M =( x)(x y x z)[~a] | 8 2 $ 2 where ~a is any assignment sending the variable y to a and the variable z to b). This means that a M = b M. \ \ Since M is transitive (in V ), every member of a or of b is a member of M. It follows that a M = a and b M = b and \ \ therefore a = b. Hence M = a = b. | In sum, M thinks that for all y, z, if y and z have the same elements, then they are equal. In other words, M = Axiom of Extensionality. | ⇤ Also: Lemma Suppose M is a transitive set or a transitive proper class which is closed under unordered pairs (meaning that for all a, b M, 2 a, b M). Then M = Axiom of Unordered pairs. { }2 | Proof. Let c = a, b M. Check, as in the previous proof, that { }2 M =( x)x c x = a x = b. | 8 2 $ _ Similarly: Lemma Suppose M is a transitive set or a transitive proper class. Suppose a M for every a M. Then M = Union set Axiom. 2 2 | Lemma S Suppose M is a transitive set or a transitive proper class. Suppose for every a M there is some b M such that 2 2 b = (a) M. Then M = Power set axiom. P \ | [Proofs: Exercises.] Note: There are situations in which there are transitive models M of fragments of ZFC, or even of all of ZFC, and some a M 2 such that (a)M is strictly included in (a) (i.e., there are P P subsets b of a such that b / M). 2 Lemma Suppose M is a transitive set or a transitive proper class. If ! M, then M = Infinity. 2 | Proof idea: As in the previous proofs. The point is that M recognises correctly, recognises correctly that something is ; an ordinal, and recognises correctly that something is the successor of an ordinal. We say that the notion of ordinal is absolute with respect to transitive models. It is possible to identify large families of properties that are absolute with respect to transitive models by virtue of their being definable by syntactically ‘simple’ formulas (from the point of view of their quantifiers): A formula ' is restricted if all its quantifiers are restricted. In other words, they appear in subformulas of the form ( x)(x y ) and ( x)(x y ). 8 2 −! 9 2 ^ We generally abbreviate the above by ( x y) and 8 2 ( x y) , respectively. 9 2 The following general fact is not difficult. Fact Suppose '(~x) is a restricted formula and M is transitive. The following are equivalent for every ~a M<!. 2 (1) '(~a) (2) M = '(~a) | Note: The notion of finiteness is also absolute with respect to transitive models but, on the other hand, the notion of countability is highly non–absolute with respect to transitive models: There are transitive models M and a M such that 2 M = a is uncountable | but there is a bijection f : ! a, so a is countable in V . The −! problem of course is that f is not in M. We will soon see that there are transitive models of (fragments of) ZFC such that all their sets are countable in V . And even the whole model can be countable in V . Note: The notion of finiteness is also absolute with respect to transitive models but, on the other hand, the notion of countability is highly non–absolute with respect to transitive models: There are transitive models M and a M such that 2 M = a is uncountable | but there is a bijection f : ! a, so a is countable in V . The −! problem of course is that f is not in M. We will soon see that there are transitive models of (fragments of) ZFC such that all their sets are countable in V . And even the whole model can be countable in V . The notion of choice function is also absolute with respect to transitive models: If M is transitive, a M consists of nonempty 2 sets and f M, then f is a choice function for a if and only if 2 M = “f is a choice function for a”. Hence: | Lemma Let M be a transitive set or a transitive proper class. Suppose for every a M consisting of nonempty sets there is a choice 2 function f for a, f M.