Measurable Cardinals and Scott’s Theorem Chris Mierzewski Mathematical Logic Seminar 1/29 Measurable Cardinals and Scott’s Theorem From measures to cardinals Measurable Cardinals Lebesgue’s Measure Problem Is there a measure : P(R) R such that I is not the constant zero function ! I is -additive I For any sets X, Y R, if X = fy + r j y 2 Y g for some fixed r 2 R, then (X) = (Y )? Vitali (1905): No such thing. 2/29 Measurable Cardinals and Scott’s Theorem From measures to cardinals Measurable Cardinals Banach: Is there any set S admitting a measure : P(S) [0, 1] such that I (S) = 1 ! I is -additive I (fsg) = 0 for all s 2 S? For which cardinals is there a non-trivial finite measure defined over (i.e. on P())? 3/29 Measurable Cardinals and Scott’s Theorem From measures to cardinals Measurable Cardinals Ulam: If is the smallest cardinal with a non- trivial finite measure over , then 2@0 or admits a non-trivial measure that takes only values in f0, 1g. I If is the smallest cardinal with a non-trivial finite measure , then is -additive. I So we can ask: does there exist any uncountable with a f0, 1g-valued measure over it? 4/29 Measurable Cardinals and Scott’s Theorem From measures to cardinals Measurable cardinals Relation between -additive measures and non-principal, -complete ultrafilters over : U = X j (X) = 1 Definition (Measurable Cardinal) An uncountable cardinal is mesurable if there exists a non-principal, -complete ultrafilter over . 5/29 Measurable Cardinals and Scott’s Theorem From measures to cardinals Some facts about measurable cardinals Suppose is measurable, and U a non-principal, -complete ultrafilter over . Then the following hold: I If X 2 U, then jXj = . I is regular. I (Tarski–Ulam) is inaccessible. So if is measurable, then (V, 2) ZFC; thus ZFC cannot prove the existence of measurable cardinals. They are indeed large. In fact: I (Hanf–Tarski (1960)) Least inaccessible cardinal < least measurable cardinal. J.Bell: “GARGANTUAN proportions...” 6/29 Measurable Cardinals and Scott’s Theorem From measures to cardinals Scott’s Theorem Theorem (Scott) If there exists a measurable cardinal, then V =6 L. Woodin’s "Meta-Corollary" “The whole point of set theory is to study infinity. You can’t deny large cardinals. So [the statement V=L] is just not true.” 7/29 Measurable Cardinals and Scott’s Theorem From measures to cardinals Scott’s Theorem Theorem (Scott) If there exists a measurable cardinal, then V =6 L. Woodin’s "Meta-Corollary" “The whole point of set theory is to study infinity. You can’t deny large cardinals. So [the statement V=L] is just not true.” 8/29 Measurable Cardinals and Scott’s Theorem Background about L The constructible universe: Göd-L Recall: I L is the smallest inner model (in V ). That is, if M is a transitive class containing all ordinals, then L M. Aternative characterisation of L: via Gödel operations and the associated closure operator. Def(A) = cl(A [ fAg) \P(A) 9/29 Measurable Cardinals and Scott’s Theorem Background about L Alternative characterisation of L: Gödel operations I G1(X, Y ) := fX, Y g I G2(X, Y ) := X ¢ Y I G3(X, Y ) := (u, v) j u 2 X ^ v 2 Y ^ u 2 v I G4(X, Y ) := X n Y I G5(X, Y ) := X \ Y I G6(X) := S X I G7(X) := dom(X) I G8(X) := (u, v) j (v, u) 2 X I G9(X) := (u, v, w) j (v, w, v) 2 X I G10(X) := (u, v, w) j (v, w, u) 2 X 10/29 Measurable Cardinals and Scott’s Theorem Background about L Alternative characterisation of L: Gödel operations We can define: L0 := ; L +1 := cl(L [ fL g) \P(L ) [ L := L 2On Theorem (Gödel) A transitive class M is an inner model of ZF if and only if I M is closed under Gödel operations I M is almost universal (whenever X M, then X Y for some Y 2 M. 11/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof Scott’s Proof I Take , the least measurable cardinal, and U a non-principal, -complete ultrafilter over . I Build an ultrapower (V /U, 2U ) of V , and collapse it to form a transitive class model (M, 2). I Using a variant of Łoś’ Theorem, there is an elementary embedding : V M. I Using V = L, show that M = V , so that is an elementary embedding ! of V . (V , 2) (M, 2) id (V /U, 2U ) 12/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof Scott’s Proof I Using the -completeness of U, show that (i) for any < , ( ) < , and (ii) we have () > . I By Łoś’ Theorem, we have that (M, 2) ‘() is the least measurable cardinal’, and since M = V , () is the least measurable cardinal in V . I But we have shown that () > , which contradicts the minimality of (). I Contradiction. If V = L, there can be no measurable cardinals. 13/29 The ultrapower V /U is the class of minimal-rank representatives of equivalence classes (Scott’s trick): [f]:= g2 V f ∼U g and 8h 2 V , if f ∼U h then rk(g) rk(h) Then each[f] is a set. I Thus we get a proper class V /U := [f] f : V ! Measurable Cardinals and Scott’s Theorem Scott’s Proof Ultrapowers of proper classes: Scott’s trick I We want to build ultrapowers of proper class models. Let a cardinal and U an ultrafilter over . For any f, g 2 V , we let f∼U g if and only if f < j f( ) = g( )g 2 U 14/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof Ultrapowers of proper classes: Scott’s trick I We want to build ultrapowers of proper class models. Let a cardinal and U an ultrafilter over . For any f, g 2 V , we let f∼U g if and only if f < j f( ) = g( )g 2 U The ultrapower V /U is the class of minimal-rank representatives of equivalence classes (Scott’s trick): [f]:= g2 V f ∼U g and 8h 2 V , if f ∼U h then rk(g) rk(h) Then each[f] is a set. I Thus we get a proper class V /U := [f] f : V ! 14/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof Ultrapowers of proper classes We have V /U = [f] f 2 V The ultrapower structure comes with a membership relation 2U , defined as [f] 2U [g] if and only if < f( ) 2 g( ) 2 U We obtain the ultrapower structure (V /U, 2U ). Useful notation: write jjf 2 gjj := f < j f( ) 2 g( )g. In general, for any first order formula '(x1, ..., xn), write '(f1, ..., fn) := < '(f1( ), ..., fn( )) . 15/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof Łoś’ Theorem For any L2-formula ', and any f1, ..., fn 2 V , we have: (V , 2U ) '[f1], ..., [fn] if and only if < V , 2 'f , ..., f 2 U ( ) 1( ) n( ) (Equivalently, iff jj'(f1, ..., fn)jj 2 U.) 16/29 I We will show that V can be elementarily embedded in (V /U, 2U ), and (V /U, 2U ) can be collapsed to a transitive class model (M in),giving an elementary embedding of (V , 2) into (M, 2). (V , 2) id (V /U, 2U ) Measurable Cardinals and Scott’s Theorem Scott’s Proof I Assume V = L, and assume there is a measurable cardinal. Let be the least measurable cardinal. I Take the ultrapower (V /U, 2U ), where U is a non-principal, -complete ultrafilter on . 17/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof I Assume V = L, and assume there is a measurable cardinal. Let be the least measurable cardinal. I Take the ultrapower (V /U, 2U ), where U is a non-principal, -complete ultrafilter on . I We will show that V can be elementarily embedded in (V /U, 2U ), and (V /U, 2U ) can be collapsed to a transitive class model (M in),giving an elementary embedding of (V , 2) into (M, 2). (V , 2) id (V /U, 2U ) 17/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof I Assume V = L, and assume there is a measurable cardinal. Let be the least measurable cardinal. I Take the ultrapower (V /U, 2U ), where U is a non-principal, -complete ultrafilter on . I We will show that V can be elementarily embedded in (V /U, 2U ), and (V /U, 2U ) can be collapsed to a transitive class model (M, 2), giving an elementary embedding of (V , 2) into (M, 2). (V , 2) (M, 2) id (V /U, 2U ) 18/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof I Assume V = L, and assume there is a measurable cardinal. Let be the least measurable cardinal. I Take the ultrapower (V /U, 2U ), where U is a non-principal, -complete ultrafilter on . I We will show that V can be elementarily embedded in (V /U, 2U ), and (V /U, 2U ) can be collapsed to a transitive class model (M, 2), giving an elementary embedding of (V , 2) into (M, 2). (V , 2) (M, 2) id (V /U, 2U ) 19/29 Measurable Cardinals and Scott’s Theorem Scott’s Proof Embedding (V , 2) (M, 2) id (V /U, 2U ) I For any set x, let cx : V be the constant map cx : 7 x. I The map id : V V /U is defined as id(x) = [cx] ! ! ' 2 L By Łoś, for any !2 we have : (V , 2) ' a1, ..., an iff (V /U, 2U ) ' [ca1 ], ..., [can ] so that id is an elementary embedding of V into V /U.