Proof theory for set theories

Transcription

1 Proof theory for set theories Toshiyasu Arai (Chiba, Japan) Aug.7, LC2015, Helsinki, Finland 1

2 Here we are concerned with formal proofs in set theory. Which kind of sets are proved to exist? 1. Bounding on provably existing countable ordinals in, e.g., ZF. 2. Proof-theoretic reductions of, e.g., the existence of a weakly compact cardinal K in terms of iterations of the Mahlo operations on K. 2

3 PLAN of TALK 1. Axioms of set theory, pp Ordinal analyses, p Skolem hulls, pp Hydra games for recursively regular ordinals, pp Lifting the ordinal analyses to ω 1,pp Hydra games for uncountable regular ordinals, pp

4 1 Axioms of set theory Axioms of Zermelo-Fraenkel set theory ZF: foranyfirst-orderformulas ϕ and θ Extensionality z(z x z y) x = y Empty empty set exists Pair unordered pair {x, y} exists Union union x = {z : y x(z y)} exists Infinity ω = {x : x y x(y {y} x)} exists Foundation x( y x ϕ(y) ϕ(x)) xϕ(x) n Σ n-separation {x a : θ(x)} exists n Σ n-collection x a y θ(x, y) c x a y c θ(x, y) Power P(a) ={x : x a} exists 4

5 Kripke-Platek set theory with the axiom of Infinity KPω is obtained from ZF-Power by restricting θ to Σ 0 -formulas(bounded formulas), which are absolute: for θ Σ 0 Extensionality z(z x z y) x = y Empty empty set exists Pair unordered pair {x, y} exists Union union x = {z : y x(z y)} exists Infinity ω = {x : x y x(y {y} x)} exists Foundation x( y x ϕ(y) ϕ(x)) xϕ(x) Σ 0 -Separation {x a : θ(x)} exists Σ 0 -Collection x a y θ(x, y) c x a y c θ(x, y) 5

6 KPω is a recursive analogue to ZF-Power. Let L = α L α be the Gödel s constructible universe. L 0 =, L α+1 is the set of all definable subsets of L α and L λ = α<λ L α for limit ordinals λ. For any uncountable regular ordinal κ > ω, α < κ f α κ(sup β<α f(β) < κ) L κ = ZF-Power While for any recursively regular ordinal κ, i.e.,κ > ω and α < κ f α κ Σ 1 (L κ )(sup β<α f(β) < κ) L κ = KPω 6

7 2 Ordinal analyses The proof-theoretic ordinal T of recursive theory T ACA 0 : T := sup{otyp( ) :T WO[ ] forrec.wo on ω} T < ω1 CK iff TisΠ 1 1-sound ( Π 1 1 ϕ[t ϕ N = ϕ]). For set theories T containing (a fragment of) KPω, T is equal to the Σ 1 (ω1 CK )-ordinal T Σ1 (ω1 CK ) of T: T = T Σ1 (ω CK 1 ) =min{α ωck 1 : Σ 1 ϕ(t ϕ L ω 1 CK L α = ϕ)} The ordinal is the limit of recursive ordinals whose existence is provable in T. 7

8 2.1 Skolem hulls Let F be a countable set of ordinal functions f : On n On. Assume that 0-ary functions 0, ω 1 belong to F. Cl(α; F) denotestheskolem hull of α under the functions in F. Proposition 2.1 Cr(F) :={β < ω 1 : Cl(β; F) ω 1 β} is club in ω 1. Let us throw the least β 0 Cr(F) inf, andthentheleast β 1 Cr(F 1 )isaddedtof 1 = F {β 0 },andsoon. 8

9 Define sets Cl α (β; F) andordinalsψ ω1 (α; F) bysimultaneous recursion on ordinals α as follows. Cl α (β; F) = Cl(β; F {ψ ω1 ( ; F) α}) ψ ω1 (α; F) = min{β ω 1 : Cl α (β; F) ω 1 β} ψ ω1 (α; F) < ω 1 is the α-th collapse of ω 1 down to countables. If α < γ, thencl α (β; F) Cl γ (β; F), ψ ω1 (α; F) ψ ω1 (γ; F) and Cr α (F) Cr γ (F) :={β < ω 1 : Cl γ (β; F) ω 1 β}. {Cr α (F)} α is a decreasing sequence of club subsets of ω 1,and Cr ω1 (F) isthediagonalintersectionof{cr α (F)} α<ω1. 9

10 For F ω = {0, ω 1 } {λxy.x + y, λx.ω x },letuswrite ψ ω1 (α) :=ψ ω1 (α; F ω ). ψ ω1 (ε ω1 +1) isarecursiveordinal< ω1 CK (Howard ordinal), and the proof-theoretic ordinal of KPω, ψ ω1 (ε ω1 +1) = KPω. The same holds for, e.g., the Kripke-Platek set theory KP 2 above L ω CK and ψ ω1 (ε ω2 1 +1), where Skolem hulls under F 2 = {0, ω 1, ω 2 } {λxy.x + y, λx.ω x } are considered with two collapsing functions ψ ωi (α; F 2 )=min{β ω i : Cl α (β; F 2 ) ω i β} for i =1, 2. 10

11 2.2 Hydra games for recursively regular ordinals [Kirby-Paris82] introduced a hydra game, whose termination is independent from the first order arithmetic PA.Then[Buchholz87] extended it to an independence result for the theories ID ν (ν ω) of ν-times iterated positive inductive definitions over PA, where ID 1 is proof-theoretically equivalent to KPω, andid 2 to KP 2. Let us modify the hydra game of Buchholz. Our modification is not elegant, but easy to extend it to set theories. Each hydra is a term over symbols {0, +,D 0,D 1,D 2 } { }. 1 := D 0 (0), ω 1 := D 1 (0), and ω 2 := D 2 (0). For a 0, D 0 (a),d 1 (a) denotecollapsingfunctionsψ ω1 (a), ψ ω2 (a), resp. 11

12 Definition 2.2 (Simultaneous inductive definition of H and Tm exp.) 1. 0 H and {a 0,...,a n } H (a a n ) H. 2. a H,t Tm exp a t H. 3. a H D i (a) D i H. 4. Tm exp is the closure of {0, 1, ω 1 } D 0 D 1 under + and a ω a where ω a = D 2 (a) fora 0. Each hydra a denotes a unique ordinal when + are replaced by the natural (commutative) sum #, D 0 (a) byψ ω1 (a), D 1 (a) by ψ ω2 (a) fora 0,D 1 (0) by ω 1,andD 2 (0) by ω 2. 12

13 H i := H ω 1+i. Let v(t) denotethevalueof(closed)term t Tm exp. Definition 2.3 For each hydra a H, domain dom(a) { (= 0), 1(= {0}), ω, H i (i =0, 1)} {multi t : t Tm exp } and hydras a[z] H for z dom(a) aredefined. ([ ].1) dom(0) =. ([ ].2) dom(1) = 1, 1[0] = 0 where 1 = D 0 (0). ([ ].3) dom(d i+1 (0)) = H i ; (D i+1 (0))[z] =z for i =0, 1. ([ ].4) a =(a a k )(k>0): dom(a) =dom(a k ); a[z] = (a a k 1 + a k [z]). 13

14 ([ ].5) Let a = D i (b) withb 0. ([ ].5.1) If b = b 0 +1, then dom(a) =ω; a[n] =(D i (b 0 )) 2. (D i (b 0 +1))[n] =D i (b 0 ) n in [Kirby-Paris82]. ([ ].5.2) If dom(b) {ω, H j : j<i}, then dom(a) =dom(b); a[z] =D i (b[z]). ([ ].5.3) If dom(b) {H j : j i}, then dom(a) =ω. Then let l = D i (b[1]), and r = D i (l + b[1]). Let a[n] :=l + r. (D i (b))[n] =D i (b[d j (b[1])]) in [Buchholz87]. It remains to consider a multiplication type a t for terms t Tm exp {ω 2 }. 14

15 ([ ].6) dom(a t) =multi t,2 := {s : s Tm exp, v(s) <v(t)}; (a t)[s] =(a s)+a[0] where (a 0) + b := b. ([ ].5.4) Let dom(b) =multi t,2 with v(t) > 0. (D 1 (b))[s] = D 1 (b[s]) for s dom(d 1 (b)) = multi t,1 = {s multi t,2 : D 1 (s) C 1 (b)}, where D i (s) thesetofconstantsind i used in building s and C i (b) thesetofcomponentsofb in D i. ([ ].5.5) Let dom(b) = multi t,i with v(t) > 0andi = 1, 2. dom(d 0 (b)) = ω. a[n] :=D i (b[s n ]) for a term s n whose value is maximal in the finite set {s multi t,i : s 2 n, j < i(d j (s) C j (b))}. s k iff the depth and the width of s simultaneously bounded by k. 15

16 For each a H 0, dom(a) {0, 1, ω}. When dom(a) =1,let a[n] :=a[0] for any n ω. Definition 2.4 (Hydra games (battles)) Let a 0 H 0 be a given hydra. Define hydras a n H 0 recursively as follows. 1. If a n =0,thena n+1 =0. 2. a n+1 = a n [n] ifa n 0. Let (H) 2 denote the statement saying that for any a 0 H 0,thereexistsann<ω such that a n =0. 16

17 For the Kripke-Platek set theory KP 2 above L ω CK 1, Theorem KP 2 proves the statement (H) 2 for each hydra a 0 H KP 2 does not prove the statement (H) 2. Theorem follows from the Lemma 2.6 KP 2 (H) 2 CON(KP 2 ) 17

18 3 Lifting the ordinal analyses to ω 1 Let us extend Cl(α; F ω )totheσ 1 -Skolem hull Hull σ Σ 1 (α {ω 1 }) of α {ω 1 } on L σ ω 1. This gives a characterization of the regularity of the ordinal ω 1. Definition 3.1 For X L σ and a L σ, a Hull σ Σ 1 (X) {a} is Σ 1 (X)-definable on L σ. Definition 3.2 (Mostowski collapsing function F Σ 1 X ) (F Σ 1 X ) 1 : L γ = Hull σ Σ1 (X) Σ1 L σ for an ordinal γ σ Let us denote, though σ dom(f Σ 1 X )=Hullσ Σ 1 (X), F Σ 1 X (σ) :=γ. 18

19 Fact. Letω α < κ < σ with α = ω ωα0 for some α 0, κ alimit ordinal, and L σ = KPω +(Π 1 -Collection). Then the following conditions are mutually equivalent: 1. α κ L σ L κ. 2. L σ = α <cf(κ). 3. For the Mostowski collapse F Σ 1 x {κ} (y) thereexistsacritical point x such that α <x<κ and Hull σ Σ 1 (x {κ}) κ x and F Σ 1 x {κ} (σ) < κ In what follows let κ = ω 1 and write ω 2 for the least ordinal σ > ω 1 such that L σ = KPω +(Π 1 -Collection). 19

20 Skolem hulls H α (β) withstrongerclosurepropertiesaredefined through critical points Ψ ωi (α) fori =1, 2. Definition 3.3 H α (β) isaskolemhullof{0, ω 1, ω 2 } β under the functions +, α ω α, Ψ ωi α,theσ 1 Skolem hulling: Y Hull ω 2 Σ 1 (Y ω 2 ) and the Mostowski collapsing functions (x = Ψ ω1 (γ), δ) F Σ 1 x {ω 1 } (δ) (x = Ψ ω2 (γ), δ) F Σ 1 x (δ). Ψ ωi (α) =min{β ω i : H α (β) ω i β}. 20

21 Let (ω 1 )denoteanaxiomstatingthat thereexistsanuncountable regular ordinal, and T 1 := KPω +(V = L)+(Π 1 -Collection) + (ω 1 ). For each n<ω, T 1 α < ω n (ω 2 +1) x <ω 1 (x = Ψ ω1 (α)) with a Σ 2 -formula x = Ψ ω1 (α). Conversely Theorem 3.4 ([A 1]) For a sentence x L ω1 ϕ(x) withaσ 2 -formula ϕ(x), if T 1 x L ω1 ϕ(x) then n <ω[t 1 x L Ψω1 (ω n (ω 2 +1))ϕ(x)]. 21

22 3.1 Hydra games for uncountable regular ordinals Each hydra is a term over symbols {0, +,D 0,D 1,D 2 } { } F µ F F where F µ = {f A : A 0 } with the µ-operator f A on ordinals = f A (x 1,...,x n ) { min{d On : A(d; x1,...,x n )} if xa(x; x 1,...,x n ) 0 otherwise and F F = {F x : β(x = Ψ ω2 (β))} {F x {ω1 } : β(x = Ψ ω1 (β))}. For a 0,D 0 (a),d 1 (a)denotecollapsingfunctionsψ ω1 (a), Ψ ω2 (a). 22

23 The set H µ (F 0 )ofhydrasandthesettm µ (F 0 )oftermsovera finite set F 0 F µ of function symbols are defined simultaneously as in Definition 2.2 with the following clauses: 2. a H µ (F 0 ),t Tm µ (F 0 ) a t H µ (F 0 ). 4. Tm µ (F 0 )istheclosureof{0, 1, ω 1 } D 0 D 1 under +, a ω a = D 2 (a), f A F 0 and F where F : (x, t) F x (t),f x {ω1 }(t). Each hydra a denotes again a unique ordinal. 23

24 As in Definition 2.3, dom(a), a[z] H µ (F 0 )aredefinedwith the following clauses for µ-terms s. ([ ].6) Let t Tm µ (F 0 ) {ω 2 }. dom(a t) =multi t,2 := {s : s Tm µ (F 0 ), v(s) <v(t)}; (a t)[s] =(a s)+a[0]. ([ ].5.4) Let dom(b) =multi t,2 with v(t) > 0. (D 1 (b))[s] = D 1 (b[s]) for s dom(d 1 (b)) = multi t,1 = {s multi t,2 : D 1 (s) C 1 (b)}. ([ ].5.5) Let dom(b) = multi t,i with v(t) > 0andi = 1, 2. dom(d 0 (b)) = ω. a[n] :=D i (b[s n ]) for a term s n whose value is maximal in the finite set {s multi t,i : s 2 n, j < i(d j (s) C j (b))}. 24

25 Definition 3.5 (Hydra games (battles)) Let a 0 H 0 (F 0 )beagivenhydra. Definehydrasa n H 0 (F 0 ) recursively as follows. 1. If a n =0,thena n+1 =0. 2. a n+1 = a n [n] if a n 0, where the RHS is determined from F 0. Let (H) µ denote the statement saying that for any finite F 0 F µ and any a 0 H 0 (F 0 ), there exists an n<ω such that a n =0. 25

26 For the set theory T 1 = KPω +(V = L)+(Π 1 -Collection)+(ω 1 ) Theorem T 1 proves the statement (H) µ for each hydra a 0 H 0 (F 0 ). 2. T 1 does not prove the statement (H) µ. Theorem follows from a consistency proof of T 1. Lemma 3.7 T 1 (H) µ CON(T 1 ) The second hydra game looks the same as the first one. Then where the power of the second lies in? 26

27 Partly due to a richness of arms of the second. Terms in the first Tm exp are generated only by + and ω t,while µ-operators f A (and Mostowski collapses) are available for the second Tm µ. The crux is the fact that in the second, the relation Ψ ω1+i (a) =D i (a) <D j (b) =Ψ ω1+j (b) isfarfromdecidable. In the first battle, Hercules can be a machine, and he needs only to chop off the (rightmost) head of hydra. In the second, it is easy for Hydra to deceive Hercules by replacing c D i (a) byabigger one c D j (b) +c[0]. Hercules needs to be so intellectual to beat the clever Hydra. 27

28 Thank you for your attention! 28

29 1. Bounding on provably existing countable ordinals in, e.g., ZF. α ZF = ZF +(V = L) ω1 := inf{α ω 1 : ϕ[zf +(V = L) x L ω1 ϕ x L α ϕ]} It is obvious that such a countable ordinal α ZF < ω 1 exists since there are only countably many formal proofs. But the definition of α ZF tells us nothing about how it looks like. 29

30 Theorem 3.8 ([A14].) α ZF = ZF +(V = L) ω1 = inf{α ω 1 : ϕ[zf +(V = L) x L ω1 ϕ x L α ϕ]} = Ψ ω1 (ε I+1 ):=sup{ψ ω1,n(ω n (I +1)):n<ω} I denotes the least weakly inaccessible cardinal, and Ψ ω1,n(α) < ω 1 is a collapsing function, which is defined through iterations of Mostowski collapsings. 30

31 2. Proof-theoretic reductions of higher indescribability to iterations of lower indescribabilities, e.g., reduction of the existence of a weakly compact cardinal K to the assumptions that the Mahlo operations M can be iterated on K. κ M(X) iff X κ is stationary in κ iff X meets every club subset of κ. A Σ n+1 -class Mh ξ n for ordinals ξ is defined through iterations of Mostowski collapsings and Mahlo operations M such that for the least weakly inaccessible cardinal I>K Theorem 3.9 ([A13]) ZF +(V = L)+(K is weakly compact) is Σ 1 2(K)-conservative over ZF +(V = L)+{K Mh ω n(i+1) n : n<ω}. 31

32 References [A91] T. Arai, A slow growing analogue to Buchholz proof, Ann. Pure Appl. Logic 54 (1991), [A97], A sneak preview of proof theory of ordinals, Ann. Japan Assoc. Philos. Sci. 20 (2012), [A98], Consistency proof via pointwise induction, Arch. Math. Logic 37 (1998), [A03], Proof theory for theories of ordinals I:recursively Mahlo ordinals, Ann. Pure Appl. Logic 122 (2003) [A13], Proof theory of weak compactness, Jour. Math. Logic 13 (2013), [A14], Lifting proof theory to the countable ordinals: Zermelo-Fraenkel s set theory, Jour. Symb. Logic 79 (2014), [A 1], ω 1 under Π 1 -Collection,draft. [A 2], Hydra games for regular ordinals, draft. [Buchholz87] W. Buchholz, An independence result for (Π 1 1 CA)+BI, Ann. Pure Appl. Logic 33 (1987), [Kirby-Paris82] L. Kirby and J. Paris, Accessible independence results for Peano Arithmetic, Bull. London Math. Soc. 14 (1982),