Consistency of Strictly Impredicative NF and a little more... - PDF document

Consistency of Strictly Impredicative NF and a little more... Sergei Tupailo Centro de Matem´ atica e Aplica¸ c˜ oes Fundamentais Universidade de Lisboa sergei@cs.ioc.ee Tallinn, January 26, 2012 Exposition of the paper S. Tupailo. Consistency of Strictly Impredicative NF and a little more... Journal of Symbolic Logic 75(4), 1326–1338, 2010 L ∈ := { = , ∈} . Extensionality is an axiom � � Ext : ∀ x ∀ y ∀ z ( z ∈ x ↔ z ∈ y ) → x = y . Definition 1 Stratification of a formula ϕ is an assignment of natural numbers (type indices) to variables (both free and bound) in ϕ s.t. for atomic subformulas of ϕ only the following variants are allowed: (a) x i = y i ; (b) x i ∈ y i +1 . 1/23

A formula ϕ is stratified iff there exists a stratification of ϕ . Equivalently, a formula is stratified iff it can be obtained from a formula of Simple Type Theory by erasing type indices (and renaming variables if necessary). Examples. The formula x ∈ y ∧ y ∈ z is stratified, but the formula x ∈ y ∧ y ∈ x is not. Stratified Comprehension is an axiom scheme � � SCA : ∃ y ∀ x x ∈ y ↔ ϕ [ x ] , for every stratified formula ϕ with y not free in ϕ . NF := SCA + Ext . V does exist: V := { x | x = x } . So, V ∈ V , V = P ( V ), etc. Foundation fails, Cantor’s Theorem fails, as well as many other ZFC theorems, too. Known facts: • Consis( NF + . . . ) → Consis( ZF + . . . ); • NF ⊢ ¬ AC ; • NF ⊢ Inf ; • PA ⊢ Consis( NF 3 ); • NF = NF 4 ; • . . . 2/23

Main unknown question (since 1937): • Consis( ZF + . . . ) → Consis( NF ) ? [2] M. Crabb´ e. On the consistency of an impredicative subsys- tem of Quine’s NF. Journal of Symbolic Logic 47, pp. 131–136, 1982. Definition 2 (Crabb´ e) An instance of Stratified Comprehen- sion � � SCA : ∃ y ∀ x x ∈ y ↔ ϕ [ x ] , (1) is predicative iff there is a stratification of (1) s.t. the indices of bound variables in ϕ are < type( y ) , and the indices of free variables in ϕ are ≤ type( y ) . NFP is a subsystem of NF where SCA is restricted to pred- icative instances. NFI (”mildly impredicative”) is an extension of NFP which allows bound variables in ϕ of types ≤ type( y ). Theorem 3 ([Crabb´ e 82]) Both NFP and NFI are consis- tent, where in addition | NFP | < | EA | , | PA 2 | ≤ | NFI | < | PA 3 | . Two kinds of proofs: model-theoretic (countably saturated mod- els) and proof-theoretic (cut-elimination) . Theorem 4 ([Holmes 99]) | NFI | = | PA 2 | . 3/23

Consider the Union axiom: � � U : ∀ z ∃ y ∀ x x ∈ y ↔ ∃ v ( v ∈ z ∧ x ∈ v ) . (2) Note that U is in NF , but not in NFI : x ∈ y ↔ ∃ v 1 ( v ∈ z ∧ x ∈ v ) ∀ z 2 ∃ y 1 ∀ x 0 � � . Theorem 5 ([Crabb´ e 82]) NFP + U = NFI + U = NF . Definition 6 (S.T.) An instance of Stratified Comprehension � � SCA : ∃ y ∀ x x ∈ y ↔ ϕ [ x ] , is strictly impredicative iff there is a stratification of it s.t. the indices of all variables in ϕ are ≥ type( y ) − 1 . Let NFSI denote a subsystem of NF where SCA is restricted to strictly impredicative instances. Then: Theorem 7 (S.T., 08) NFSI (and a little more, e.g. exis- tence of Frege natural numbers) is consistent, too. The proof uses a bit of Model Theory, and a lot of Set Theory (forcing). 4/23

Theorem 8 ([Specker 62]) 1. NF is consistent iff there is a model of TNT [ TST is fine] with a type-shifting automorphism [=: tsau] σ . 2. NF is equiconsistent with the Theory of Types, TNTA [ TSTA is fine] with the Ambiguity scheme, Amb , ϕ ↔ ϕ + , [ ϕ + is the result of raising all type for all sentences ϕ . indices in ϕ by 1.] Proof. See [6]. ✷ Specker’s proof generalizes immediately to subsystems of NF where SCA is restricted. For NFSI , an equivalent Type Theory is Ext plus Amb plus all instances of ∃ y i +1 ∀ x i � � x ∈ y ↔ ϕ [ x ] , where all indices in ϕ are ≥ i . 5/23

From the outset, we assume consistency of ZFC . Let � M, ∈� be an Ehrenfeucht-Mostowski model of ZF + V = L , i.e. a countable model with a non-trivial external ∈ -automorphism σ . W.l.o.g.w.m.a. that σ moves up at least one regular cardinal κ (in the sense of M ): In M , sets can be enumerated by ordinals, i.e. there is a formula ϕ ( x, α ) s.t. the sentence ” ϕ gives a (class) bijection between V and On ” is true in M . By Ehrenfeucht-Mostowski, σ ( x ) � = x for some x ∈ M . Since we have a definable bijection, σ ( α ) � = α for some ordinal α ∈ M . If α < σ ( α ), fine; if not, take σ − 1 . In order to move up a cardinal, use a definable bijection α �→ ℵ α . In order to move up a regular cardinal, use a definable injection α �→ ℵ α +1 . By default, we will use forcing machinery (original results due to P. Cohen and R. Solovay) as laid out in [5] K. Kunen. Set Theory. An Introduction to Indepen- dence Proofs. Elsevier, 1980. Given a finite set S of TSTA -axioms, let n ≥ 2 be such that all indices i in S fall under 0 ≤ i ≤ n . For 0 ≤ i < n , let P i := Fn( σ i +1 ( κ ) , 2 , σ i ( κ )) (Cohen’s poset), where I Fn( κ 1 , 2 , κ 0 ) := { p || p | < κ 0 ∧ p is a function ∧ dom( p ) ⊂ κ 1 ∧ ran( p ) ⊂ 2 } (3) P n := � (see VII 6.1), and I P := I 0 ≤ i<n I P i . Note first that σ acts as a bijection between σ i ( κ ) and σ i +1 ( κ ). Let G 0 be I P 0 -generic over M . Then bi M [ G 0 ] | = ∃ h 0 h 0 : σ ( κ ) �→ P ( κ ) . 6/23

Definition 9 P <ω ( b ) := { a ⊂ b | | a | < ω } . Let g 0 ∈ M be such that bi �→ P <ω ( κ ) . g 0 : κ Defining g i := σ i ( g 0 ), we get bi g i : σ i ( κ ) �→ P <ω ( σ i ( κ )) . (4) bi Lemma 10 Given M [ G 0 ] ∋ h 0 : σ ( κ ) �→ P ( κ ) and M ∋ g 0 : bi bi κ �→ P <ω ( κ ) , there exists a bijection M [ G 0 ] ∋ f 0 : σ ( κ ) �→ P ( κ ) satisfying f 0 ↾ κ = g 0 . Proof. Work in M [ G 0 ]. Since |P ( κ ) | = σ ( κ ), |P <ω ( κ ) | = κ and P ( κ ) = P <ω ( κ ) � P ≥ ω ( κ ), we must have |P ≥ ω ( κ ) | = σ ( κ ), i.e. there is a bijection h 1 between σ ( κ ) and P ≥ ω ( κ ). Now, for a ∈ P ( κ ), define f ′ 0 ( a ) by � g − 1 0 ( a ) if a ∈ P <ω ( κ ), f ′ 0 ( a ) := (5) κ + h − 1 1 ( a ) otherwise. We claim that f ′ 0 is a special bijection between P ( κ ) and σ ( κ ): (i) f ′ 0 ( a ) < σ ( κ ) is seen from (5) and the fact that σ ( κ ) is an additive principal number, i.e. an ordinal closed under ordinal sum; 0 ( a ) = g − 1 (ii) f ′ 0 is onto: if α < κ , then by the first line of (5) f ′ 0 ( a ) = α for some a ∈ P <ω ( κ ); otherwise, α = κ + β for some β < σ ( κ ), and then 0 ( a ) = κ + h − 1 f ′ 1 ( a ) for some a ∈ P ≥ ω ( κ ); 0 is 1-1 follows from (5) and the fact that both g − 1 and h − 1 (iii) f ′ are 1-1; 0 1 0 ↾ P <ω ( κ ) = g − 1 (iv) further, from the first line of (5) we have f ′ 0 . From (i-iv) above, f 0 can be taken as the inverse of f ′ 0 . ✷ bi Choose f 0 : σ ( κ ) �→ P ( κ ) as guaranteed by Lemma 10. 7/23

P 0 be a name for f 0 , so that Let τ ∈ M I bi M [ G 0 ] | = τ G 0 : σ ( κ ) �→ P ( κ ) . (6) By the Forcing Theorem VII 3.6 � M � bi ˇ ) ∃ p ∈ G 0 p � − ∗ ˇ �→ P (( κ ) I P 0 τ :( σ ( κ )) I . (7) P 0 P 0 I Taking p ∈ G 0 from (7) and applying σ i to this formula, we obtain � M � P i σ i ( τ ):( σ i +1 ( κ )) I bi σ i ( p ) � �→ P (( σ i ( κ )) I ˇ ) ∗ − ˇ . (8) P i P i I Define G i +1 := σ ′′ G i , 0 ≤ i < n − 1, and G := � 0 ≤ i<n G i . Then each G i contains σ i ( p ) and is I P i -generic over M – see Lemma 11. It’s easily verified that G is a filter on I P = � 0 ≤ i<n I P i , but it was more of an issue whether G is generic. Also observe that σ i ( τ ) ∈ M I P i , for each i . Lemma 11 (See pp. 219–220) ⇒ σ ′′ G is σ ( P ) -generic over M. G is P -generic over M ⇐ Proof. ” G is a filter in P ” being equivalent to ” σ ′′ G is a filter in σ ( P )” follows from σ being an isomorphism between P and σ ( P ). For the ”generic” part, it follows from ” D is dense in P ” ⇔ ” σ ′′ D is dense in σ ( P )” ( σ isomorphism) and σ ′′ D = σ ( D ) ( σ ∈ -automorphism of M ). ✷ 8/23

Starting with the complete embeddings I P i �→ � 0 ≤ i<n I P i , define P as in VII 7.12. P i �→ M I natural embeddings ı i : M I Lemma 12 For each i , 0 ≤ i < n , M [ G i ] is a transitive sub- model of M [ G ] . Then x = ρ G i for ρ ∈ M I P i . Proof. Let x ∈ M [ G i ]. Then P and x = ρ G i = ( ı i ( ρ )) G by VII 7.13(a), so that ı i ( ρ ) ∈ M I x ∈ M [ G ]. Now, assume x = ρ G i = ( ı i ( ρ )) G ∈ M [ G i ], y = τ G ∈ M [ G ], y ∈ M [ G ] x . We need to show y ∈ M [ G i ] and y ∈ M [ G i ] x . We compute: y ∈ M [ G ] x ⇐ ⇒ τ G ∈ M [ G ] ( ı i ( ρ )) G ⇐ ⇒ ∃ p ∈ G ( � τ, p � ∈ ı i ( ρ )) P i ( � δ, p i � ∈ ρ ∧ τ = ı i ( δ ) ∃ p ∈ G ∃ p i ∈ G i ∃ δ ∈ M I VII 7.12 ⇐ ⇒ ∧ p = �∅ , . . . , p i , . . . , ∅� ) = ⇒ y = δ G i ∈ M [ G i ] ∧ δ G i ∈ M [ G i ] ρ G i . ✷ See Picture 1. Interpret variables x i of L TST n as x ∈ σ i ( κ ), and interpret x i ∈ i y i +1 as x ∈ ( σ i ( τ )) G i ( y ). First note that from (8) we have bi = ( σ i ( τ )) G i : σ i +1 ( κ ) �→ P ( σ i ( κ )) , M [ G i ] | (9) for each 0 ≤ i < n . For brevity, we denote L. 12 f i := ( σ i ( τ )) G i ∈ M [ G i ] ⊂ M [ G ] . 9/23