MAGIC Set theory. lecture 6

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1 MAGIC Set theory lecture 6 David Asperó Delivered by Jonathan Kirby University of East Anglia 19 November 2014

2 Recall: We defined (V : 2 Ord) by recursion on Ord: V 0 = ; V +1 = P(V ) V = S {V : < } if is a limit ordinal. (V : 2 Ord) is called the cumulative hierarchy. Definition For every x 2 S 2Ord V, rank(x) =min{ 2 Ord : x 2 V +1 }

3 We have seen, by induction on the ordinals, that V is transitive for every ordinal. Proposition For all <, V V. Proof. Again by induction on. This is vacuously true for = 0. For a nonzero limit ordinal, V V by definition of V. For = + 1, V = P(V ). If =, then we are done since every member of V is a subset of V (as V is transitive) and therefore V P(V ). If <, then V V by induction hypothesis. But V P(V ) by the previous case, and hence V P(V ) =V.

4 Definition For every x 2 S 2Ord V, let rank(x) be the first such that x 2 V +1. Definition For every set x, the transitive closure of x, denoted by TC(x), is S {Xn : n <!} where X 0 = x X n+1 = S X n So TC(x) =x [ S x [ SS x [ SSS x [... [Exercise: TC(x) is the least transitive set y such that x y. In other words, TC(x) = T {y : y transitive, x y}.]

5 Definition V denotes the class of all sets; that is, V = {x : x = x}. Definition WF = S {V : 2 Ord}: The class of all x such that x 2 V for some ordinal. Note: WF is a transitive class: y 2 x 2 V implies y 2 V since V is transitive.

6 Theorem (ZF) V = WF Proof: Suppose, towards a contradiction, that there is some set x such that x /2 WF. Let y = TC(x). y /2 WF: Suppose y 2 V. Since x y V (where y V is true by transitivity of V ), x 2P(V )=V +1. Contradiction. By Foundation we may find a 2 y [{y}, a 2 minimal in y [{y}, such that a /2 WF.

7 For every z 2 a, it follows that z 2 y [{y} (by transitivity of y [{y}) and therefore z 2 V for some 2 Ord by 2 minimality of a among {w 2 y [{y} : w /2 WF}. Hence, the function rank a sending z 2 a to rank(z) is defined for all z 2 a. But then range(rank a) has to be a set by Replacement and therefore there is some ordinal such that >rank(z) for every z 2 a [No set X of ordinals can be cofinal in Ord (i.e., such that for every 2 Ord there is some 2 X with < ). Why? Otherwise S X = Ord, which is not a set (Burali Forti), but S X is a set if X is a set by Union Axiom. Contradiction.] It follows that for every z 2 a there is some z 2 V V. < such that Hence a V and therefore a 2P(V )=V +1. Contradiction with a /2 WF.

8 The fact that V = WF realises the idea that a set is any collection built out of sets already built. This is known as the iterative conception of sets. Note that this conception of sets rules out such sets as V or the Russell class {x : x /2 x}: They couldn t possibly be sets since one needs to refer to the totality of sets for their definition, a totality to which they would belong if they were sets. Take for example, V. Certainly, if V is a set, then V 2 V. But this goes against the iterative conception of set, whereby a set is built up out of previously built sets.

9 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.

10 Inner models and relativization Let (M, 2 M ) be a submodel, or inner model, defined by a formula (x); in other words, M = {a : (a)} and, for all a, b 2 M, a 2 M b if and only if a 2 b (we usually leave out 2 M and write M instead of (M, 2 M )). (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 2 y) M is x 2 y. (x = y) M is x = y. (' 0 _ ' 1 ) M is ' M 0 _ 'M 1. ( ') M is ' M. ((8x)('( x)) M is 8x( (x)! ' M ( x)). We may also write (8x 2 M)' M (x).

11 The following is quite obvious and can be easily proved by induction on the complexity of : Note: Suppose T is a theory in the language of set theory. Suppose (N, E) is a structure in the language of set theory, and suppose M is an inner model in N. Then (N, E) = M for every 2 T if and only if (M, E) = T. 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 2 N :(N, E) = (a)}), and we want / need to emphasise that M is the inner model defined by (x) as defined within (N, E), then we often write M N instead of M. Example: WF M Note: For every ordinal, V WF = V (here V refers to the set, definable from the parameter, with the definition that we have seen).

12 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 2 M and suppose M = (8x)(x 2 a $ x 2 b) (this of course is shorthand for M = (8x)(x 2 y $ x 2 z)[~a] where ~a is any assignment sending the variable y to a and the variable z to b).

13 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.

14 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 2 M, {a, b} 2M). Then M = Axiom of Unordered pairs. Proof. Let c = {a, b} 2M. Check, as in the previous proof, that M = (8x)x 2 c $ x = a _ x = b.

15 Similarly: Lemma Suppose M is a transitive set or a transitive proper class. Suppose S a 2 M for every a 2 M. Then M = Union set Axiom. Lemma Suppose M is a transitive set or a transitive proper class. Suppose for every a 2 M there is some b 2 M such that b = P(a) \ M. Then M = Power set axiom. [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 2 M such that P(a) M is strictly included in P(a) (i.e., there are subsets b of a such that b /2 M).

16 Lemma Suppose M is a transitive set or a transitive proper class. If! 2 M, then M = Infinity. 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). We don t need this kind of general analysis at the moment so we won t go into that now.

17 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 2 M such that 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.

18 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 2 M such that 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.

19 The notion of choice function is also absolute with respect to transitive models: If M is transitive, a 2 M consists of nonempty sets, f 2 M, and f is a choice function for M, then 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 2 M consisting of nonempty sets there is a choice function f for a, f 2 M. Then M = AC.

20 Lemma Let M be a transitive set or a transitive proper class. Suppose b 2 M whenever a 2 M and b a is definable over M, possibly from parameters (in other words, b = {c : c 2 a, M = '(c, ~p)} for some parameters ~p 2 M). Then M = Separation. Lemma Let M be a transitive set or a transitive proper class. Suppose F [a] 2 M whenever a 2 M, and F is a class function over M (in other words, if F is definable by a formula '(x, y, ~z) which, over M is functional, ~p 2 M, and a 2 M, then {c :(9b 2 a)m = '(b, c, ~p)} 2M). Then M = Replacement.

21 Our first relative consistency proof: Con(ZF \{Foundation}) implies Con(ZF) Theorem Let M = ZF \{Foundation}. Then M = WFM for every 2 ZF. Hence, WF M = ZF. Proof: By the previous lemmas and the construction of (V : 2 Ord), WF M = for every axiom of ZF \{Foundation} [go through them one by one them and check that WF is closed under the relevant operation, then apply the relevant lemma].

22 To see that M = Foundation WF holds, let us work in M: Let a 2 V, let Z a, Z 2 WF, and let b 2 Z such that rank(b) =min{rank(z) :z 2 Z }. Then WF = rank(b) =min{rank(z) :z 2 Z } by absoluteness of the relevant notions. Hence, WF thinks that the restriction of 2 to a is well founded. Since this is true for all a 2 WF, WF = Foundation

23 Corollary If ZF \{Foundation} is consistent, then ZF is consistent. Proof. Suppose ZF \{Foundation}. By the completeness theorem we may find a model M = ZF \{Foundation}. Let M 0 = WF M. By the theorem M 0 = ZF. Hence, ZF has a model and therefore it is consistent. Remark By exactly the same argument, if M = ZFC \{Foundation}, then M = WFM for every 2 ZFC. Hence, Con(ZFC \{Foundation}) implies Con(ZFC).

24 Similar relative consistency results: One can define the constructible universe L: L 0 = ; L +1 = Def (L ), where Def (L ) is the set of all subsets of L definable over L possibly with parameters, i.e., the collection of all sets of the form {b 2 L : L = '(b, a 0,...,a n 1 )} for some formula '(x, ~x) and a 0,...,a n 1 2 L. L = S < L if >0 is a limit ordinal. L is then S 2Ord L. This construction is due to Gödel. He proved that if we do this construction in ZF, then L = ZF but also L = AC and L = CH.

25 The above results imply that if ZF is consistent, then ZFC is also consistent, and in fact also ZFC+CH. Linking this to the implication we have seen we thus have that if ZF \{Foundation} is consistent, then so is ZFC+CH. These relative consistency proofs proceed by building suitable inner models. 1 Most relative consistency proofs proceed, on the other hand, by building suitable outer models of some given ground model. The construction of these outer models is done with the forcing method. This is an extremely powerful method in set theory. I will try to say something more specific about this method later on. 1 What set theorists understand by inner models are usually much more complicated than WF or L. However, the construction of L is in fact the paradigm for most of these more complicated constructions.

26 The above results imply that if ZF is consistent, then ZFC is also consistent, and in fact also ZFC+CH. Linking this to the implication we have seen we thus have that if ZF \{Foundation} is consistent, then so is ZFC+CH. These relative consistency proofs proceed by building suitable inner models. 1 Most relative consistency proofs proceed, on the other hand, by building suitable outer models of some given ground model. The construction of these outer models is done with the forcing method. This is an extremely powerful method in set theory. I will try to say something more specific about this method later on. 1 What set theorists understand by inner models are usually much more complicated than WF or L. However, the construction of L is in fact the paradigm for most of these more complicated constructions.

MAGIC Set theory. lecture 2

MAGIC Set theory. lecture 2 MAGIC Set theory lecture 2 David Asperó University of East Anglia 22 October 2014 Recall from last time: Syntactical vs. semantical logical consequence Given a set T of formulas and a formula ', we write