n -indescribabilities in proof theory Toshiyasu Arai(Chiba) 1 In - PowerPoint PPT Presentation

Π 1 n -indescribabilities in proof theory Toshiyasu Arai(Chiba) 1

In this talk let us report a recent proof-theoretic reduction on indescribable cardinals. It is shown that over ZF + ( V = L ), the existence of a Π 1 1 - indescribable cardinal is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations. The same holds for Π 1 n +1 -indescribable cardinals and Π 1 n -indescribabilities. 2

PLAN of the talk 1. Indescribable cardinals, pp. 4-10 2. Reduction of Π 1 N +1 -indescribability, pp. 11-23 3

1 Indescribable cardinals Consider the language {∈ , R } with a unary predicate symbol R . Π 1 0 denotes the set of first-order formulas in the language {∈ , R } , and Π 1 n the set of second-order formulas ∀ X 1 ∃ X 2 · · · QX n ϕ . Definition 1.1 [Hanf-Scott61] For n ≥ 0 , a cardinal κ is said to be Π 1 n -indescribable iff for any A ⊂ V κ and any Π 1 n -sentence ϕ ( R ) , if V κ | = ϕ [ A ] , then V α | = ϕ [ A ∩ V α ] for some α < κ . Definition 1.2 S ⊂ Ord is said to be Π 1 n -indescribable in κ iff for any A ⊂ V κ and any Π 1 n -sentence ϕ ( R ) , if V κ | = ϕ [ A ] , then V α | = ϕ [ A ∩ V α ] for some α ∈ S ∩ κ . 4

Facts . 1. A cardinal is inaccessible(, i.e., regular and strong limit) iff it is Π 1 0 -indescribable. 2. For regular uncountable κ , S is Π 1 0 -indescribable in κ iff S is stationary in κ , i.e., S meets every club (closed and un- bounded) subset of κ . 3. [Hanf-Scott61] A cardinal is Π 1 1 -indescribable iff it is weakly compact, i.e., inaccessible and has the tree property. By definition, κ has the tree property if every tree of height κ whose levels have size less than κ has a branch of length κ . 5

Let Rg denote the class of regular uncountable cardinals, and S ⊂ Ord . Definition 1.3 (Mahlo operation) M 0 ( S ) := { σ ∈ Rg : S is stationary in σ } = { σ ∈ Rg : S is Π 1 0 -indescribable in σ } Definition 1.4 For n ≥ 0 , M n ( S ) := { σ ∈ Rg : S is Π 1 n -indescribable in σ } . 6

Lemma 1.5 M n +1 ( Ord ) ∩ M n ( S ) ⊂ M n ( M n ( S )) . Namely if κ is a Π 1 n +1 -indescribable cardinal and S ⊂ Ord is Π 1 n -indescribable in κ , then M n ( S ) is Π 1 n -indescribable in κ . Proof . This follows from the fact that there exists a Π 1 n +1 -sentence m n ( S ) such that κ ∈ M n ( S ) iff V κ | = m n ( S ), which in turn follows from the existence of a universal Π 1 n -formula. 2 Hence if κ ∈ M n +1 ( Ord ) = M 1 n +1 , then κ ∈ M n ( M n ( Ord )) = M 2 n , M 3 n ( α < κ ) , M � n , . . . , M α n , . . . where κ ∈ M � α<κ M α n : ⇔ κ ∈ � n . 7

Actually Lemma 1.5 characterizes, over V = L , the weak com- pactness of regular uncountable cardinals κ . Theorem 1.6 [Jensen72] Assume V = L . For regular uncount- able cardinals κ , κ ∈ M 1 1 ⇔ ∀ S ⊂ κ [ κ ∈ M 0 ( S ) → Rg ∩ M 0 ( S ) ∩ κ � = ∅ ] ⇔ ∀ S ⊂ κ [ κ ∈ M 0 ( S ) → κ ∈ M 0 ( M 0 ( S ))] Theorem 1.7 [Bagaria-Magidor-Sakai ∞ ] Assume V = L . For Π 1 n -indescribable cardinals κ ∈ M 1 n , κ ∈ M 1 n +1 ⇔ ∀ S ⊂ κ [ κ ∈ M n ( S ) → M 1 n ∩ M n ( S ) ∩ κ � = ∅ ] ⇔ ∀ S ⊂ κ [ κ ∈ M n ( S ) → κ ∈ M n ( M n ( S ))] 8

Definition 1.8 Let κ be a regular uncountable cardinal. 1. S is ( − 1) -stationary in κ iff S ∩ κ is unbounded in κ . 2. λ is Π 1 − 1 -indescribable iff λ is a limit ordinal. 3. For n ≥ 0 , S is n -stationary in κ iff S meets every n -club subset of κ . 4. C is ( n + 1) -club in κ iff (a) C is n -stationary in κ , and (b) if C is n -stationary in Π 1 n -indescribable λ <κ , then λ ∈ C . 9

Let M 1 0 denotes the class of inaccessible cardinals. Proposition 1.9 [Bagaria-Magidor-Sakai ∞ ] For n ≥ 0 and κ ∈ M 1 n , κ ∈ M n ( S ) iff S is n -stationary in κ. Corollary 1.10 [Bagaria-Magidor-Sakai ∞ ] Assume V = L . For n ≥ 0 and κ ∈ M 1 n , κ ∈ M 1 n +1 iff ∀ S ⊂ κ [ S is n -stationary in κ ⇒ ∃ λ ∈ M 1 n ∩ κ ( S is n -stationary in λ )] 10

Reduction of Π 1 2 N +1 -indescribability We now ask: How far can we iterate the operation M n of Π 1 n -indescribability in Π 1 n +1 -indescribable cardinals? Or proof-theoretically: Over ZF ( ZF +(V=L)), the existence of a Π 1 n +1 -indescribable cardinal is reducible to iterations of M n ? 11

Let < ε be a ∆-predicate such that for any transitive and well- founded model V of KP ω , < ε is a canonical well ordering of type ε I +1 for the order type I of the class Ord of ordinals in V . I will show that the assumption of the Π 1 N +1 -indescribability is proof-theoretically reducible to iterations of an operation along initial segments of < ε over ZF +(V=L). The operation is a mix- ture Mh α N,n [Θ] of the operation M N of Π 1 N -indescribability and Mostowski collapsings. To define the class Mh α N,n [Θ], we need first to introduce ordinals for analyzing ZF +(V=L) proof-theoretically in [A ∞ 1]. Let I be a weakly inaccessible cardinal, and L I the set of con- structible sets of L -rank < I . 12

2.1 Skolem hulls and ZF +(V=L)-provable countable ordinals Definition 2.1 For X ⊂ L I , Hull Σ n ( X ) denotes the Σ n -Skolem hull of X in L I . a ∈ Hull Σ n ( X ) ⇔ { a } ∈ Σ L I n ( X ) ( a ∈ L I ) . Definition 2.2 (Mostowski collapsing function F ) By the Condensation Lemma we have an isomorphism (Mostowski collapsing function) F : Hull Σ n ( X ) ↔ L γ for an ordinal γ ≤ I such that F � Y = id � Y for any transitive Y ⊂ Hull Σ n ( X ) . 13

Though I �∈ dom ( F ) = Hull Σ n ( X ) write F ( I ) := γ. Let us denote the isomorphism F on Hull Σ n ( X ) ↔ L γ by F Σ n X . Given an integer n , let us define a Skolem hull H α,n ( X ) and ordinals Ψ κ,n α (regular κ ≤ I ) simultaneously by recursion on α < ε I +1 , the next ε -number above I . 14

Definition 2.3 H α,n ( X ) is a Skolem hull of { 0 , I } ∪ X under the functions + , α �→ ω α , Ψ κ,n � α (regular κ ≤ I ), the Σ n -definability: Y �→ Hull Σ n ( Y ∩ I ) and the Mostowski collapsing functions ( x = Ψ κ,n γ, δ ) �→ F Σ 1 x ∪{ κ } ( δ ) ( κ ∈ Rg ∩ I ) , ( x = Ψ I,n γ, δ ) �→ F Σ n x ( δ ) . For κ ≤ I Ψ κ,n α := min { β ≤ κ : κ ∈ H α,n ( β ) & H α,n ( β ) ∩ κ ⊂ β } . For each α < ε I +1 , ZF + ( V = L ) � Ψ κ,n α < κ . 15

Theorem 2.4 ([A ∞ 1]) For a sentence ∃ x < ω 1 ϕ ( x ) with a first-order formula ϕ ( x ) , if ZF + ( V = L ) � ∃ x < ω 1 ϕ ( x ) then ∃ n < ω [ ZF + ( V = L ) � ∃ x < Ψ ω 1 ,n ω n ( I + 1) ϕ ( x )] . Thus the countable ordinal Ψ ω 1 ε I +1 := sup { Ψ ω 1 ,n ω n ( I + 1) : n < ω } is the limit of ZF + ( V = L )-provably countable ordinals. 16

Our proof of Theorem 2.4 is based on ordinal analysis(cut- elimination in terms of operator controlled derivations in [Buchholz92]) and the following observation. Proposition 2.5 Let ω ≤ α < κ < I with α a multiplicative Then L I | principal number. = α < cf ( κ ) iff there exists an ordinal β between α and κ such that Hull Σ 1 ( β ∪ { κ } ) ∩ κ ⊂ β ( ⇔ β = F Σ 1 β ∪{ κ } ( κ )) and F Σ 1 β ∪{ κ } ( I ) < κ . 17

The class Mh α 2.2 N,n [Θ] In what follows K denotes a Π 1 N +1 -indescribable cardinal, and I the least weakly inaccessible cardinal above K . The operator H α,n ( X ) is defined as above augmented with K ∈ H α,n ( X ). In the following definition, α can be much larger than π . Definition 2.6 Let α < ε I +1 , Θ ⊂ fin ( K + 1) and K ≥ π be regular uncountable. Then π ∈ Mh α N,n [Θ] iff H α,n ( π ) ∩ K ⊂ π & α ∈ H α,n [ Θ ]( π ) & ∀ ξ ∈ H ξ ,n [ Θ ∪ { π } ]( π ) ∩ α [ π ∈ M N ( Mh ξ N,n [ Θ ∪ { π } ])] Roughly { π } in ξ ∈ H ξ ,n [ Θ ∪ { π } ]( π ) allows to define ξ from the point π . 18

For the case N = 1, i.e., Π 1 1 -indescribable cardinal K , let us examine the strength of the assumptions K ∈ Mh K +1 0 ,n [ ∅ ]. M α ( α < K + ) denotes the set of α -weakly Mahlo cardinals defined as follows. M 0 := Rg ∩ K , M α +1 = M 0 ( M α ), M λ = � { M 0 ( M α ) : α < λ } for limit ordinals λ with cf ( λ ) < K , and M λ := �{ M 0 ( M λ i ) : i < K} for limit ordinals λ with cf ( λ ) = K , where sup i< K λ i = λ and the sequence { λ i } i< K is chosen so that it is the < L -minimal such sequence. In the last case for π < K , π ∈ M λ ⇔ ∀ i < π ( π ∈ M 0 ( M λ i )). 19

Proposition 2.7 For n ≥ 1 and σ ≤ K , the followings are provable in ZF + ( V = L ) . 0 ,n [Θ] ∩ σ , and α ∈ Hull Σ 1 ( { σ , σ + } ∪ π ) ∩ 1. If σ ∈ Θ , π ∈ Mh α σ + , then π ∈ M α . 2. If σ ∈ Mh σ + 0 ,n [Θ] , then ∀ α < σ + ( σ ∈ M 0 ( M α )), i.e., σ is a greatly Mahlo cardinal in the sense of [Baumgartner-Taylor-Wagon77]. 3. The class of the greatly Mahlo cardinals below K is stationary in K if K ∈ Mh K +1 0 ,n [ ∅ ] . Proof . 2.7.3 follows from 2.7.2. 20