Strong initial segments of models of I ∆ 0 Paola D’Aquino ∗ Seconda Universita’ di Napoli Clermont-Ferrand 2006 *joint work with Julia Knight
Language L contains + , · , 0 , 1 , < PA : L -theory axiomatized by basic axioms for + and · , and induction ∀ ¯ y (( θ (0 , ¯ y ) ∧ ∀ x ( θ ( x, ¯ y ) → θ ( x + 1 , ¯ y )) → ∀ xθ ( x + 1 , ¯ y )) for any formula θ I ∆ 0 : subsystem of PA where induction is only ∆ 0 -induction ( θ ∈ ∆ 0 ). I ∆ 0 �⊢ expotentiation total (Parikh,71) exp = ∀ x > 1 ∀ y ∃ z ( x y = z ) I ∆ 0 ⊂ I ∆ 0 + exp 1
Defn [ B n formulas] The B 0 formulas are the ∆ 0 formulas. The Σ n +1 formulas have the form ( ∃ u ) ϕ , where ϕ is a B n formula. The B n +1 formulas are obtained from the Σ n +1 formulas by taking Boolean combinations and adding bounded quantifiers. Combinatorial principles are ubiquitous in arith- metical theories, e.g. Ramsey theory, pigeon- hole principle 2
Ramsey Theorem for PA : Let B be a model of PA . Let I be a cofinal definable set, and let F : I [ n ] → c = { 0 , . . . , c − 1 } be a definable partition of I [ n ] , where n is standard and c ∈ B . Then there is a cofinal definable set J ⊆ I that is homogeneous for F . Defn Let I be a subset of an L -structure A . We say that I is diagonal indiscernible for ϕ ( u, x ) if for all i < j, k in I , A | = ( ∀ u ≤ i ) [ ϕ ( u, j ) ↔ ϕ ( u, k )] . Proposition Let A be a model of PA , and let I be a cofinal definable set. For any finite r and any finite set Γ of formulas (with free variables split), there is a set J ⊆ I of size at least r that is diagonal indiscernible for all ϕ ( u, x ) ∈ Γ. Cor. In the same hypothesis get J ⊆ I cofinal definable set of diagonal indiscernible for all ϕ ( u, x ) ∈ Γ. 3
Theorem (McAloon 82) Let M be a model of I ∆ 0 . Then M has a nonstandard initial segment I which is a model of PA The proof uses diagonal indiscernibles Thm. Let A be a model of I ∆ 0 . Let I be a subset of A of order type ω such that = i < j → i 2 < j , 1. for i, j ∈ I , A | 2. I is diagonal indiscernible for all ∆ 0 -formulas. Then B = { x ∈ A : x < i for some i ∈ I } is a model of PA . We want refinements of McAloon’s result 4
Ketonen and Solovay (’81) related the follow- ing three notions: 1. α -largeness 2. Ramsey Theory 3. Wainer functions 5
α -LARGENESS ǫ 0 is the least ordinal α such that ω α = α Cantor normal form: α = ω α 1 x 1 + ω α 2 x 2 + . . . + ω α k x k where α i are ordinals, α 1 > α 2 > . . . > α k and k, x 1 , x 2 , . . . , x k ∈ ω − { 0 } Fundamental sequence: For each ordinal 0 < α < ǫ 0 , we define the x -th ordinal in the fundamental sequence { α } ( x ) as follows α = β + 1, { α } ( x ) = β , for all x α = ω β +1 , { α } ( x ) = ω β · x α = ω β , where β is a limit ordinal, { α } ( x ) = ω { β } ( x ) α = ω β · ( a + 1), where a � = 1, { α } ( x ) = ω β · a + { ω β } ( x ) α with Cantor normal form ending in ω β · a , say α = γ + ω β · a , { α } ( x ) = γ + { ω β · a } ( x ) 6
Sequence of ordinals cofinal in ǫ 0 ω 0 = 1 and ω n +1 = ω ω n Sommer in 95 formalizes the whole theory of ordinals below ǫ 0 in I ∆ 0 , including the notion of fundamental sequence, Cantor normal form, ω n ’s Defn: A sequence J = x 1 < x 2 < . . . is α - large if there is (a code for) a computation sequence C = < c 0 , c 1 , . . . , c 2 r +2 > where c 2 i is a decreasing sequence of ordinals, c 2 i +1 = x i ∈ J and c 0 = α , c 1 = x 0 , and c 2( i +1) = { c 2 i } ( c 2 i +1 ) and c 2 r +2 = 0. Notation: ( J, C ) α Example: The set X = { 3 , 4 , 5 , 6 } is ω -large, ω (3 , 4 , 5 , 6) = 3(4 , 5 , 6) = 2(5 , 6) = 1(6) = 0( ∅ ) = 0 giving the computation sequence C = < ω, 3 , 3 , 4 , 2 , 5 , 1 , 6 , 0 > . Sommer has α -largeness in I ∆ 0 + exp 7
Properties: 1. If J is α -large, where α = ω β 1 x 1 + . . . + ω β n x n then J = J n ˆ · · · ˆ J 1 , where J i is ω β i x i -large. 2. If ( J, C ) ω α then there exists J ′ ⊆ J and C ′ such that ( J ′ , C ′ ) α If C ′ = ( α, x 0 , β 1 , x 1 , . . . , β r − 1 , x r , 0) then there is a subsequence C ′′ of C C ′′ = ( ω α , x 0 , ω β 1 , x 1 , . . . , ω β r − 1 x r , ω 0 ), 3. If ( J, C ) ω α · x then for all y < x , the ordinal ω α · y appears in C . 4. If J is ω n +2 -large, with first element ≥ c , then there exists J ′ ⊆ J that is ( ω n +1 + ω 3 + c + 3)-large. 5. ( J, C ) α and j 0 1st element of J then for all x ≤ j 0 there is i s.t. { α } ( x ) = α 2 i x -unwinding of α 6. If J is cofinal definable then J is α -large for all α < ǫ 0 8
Connections with Ramsey theory by Ketonen and Solovay Thm (Inductive lemma) Let n ≥ 1 and let ω ≤ α < ǫ 0 . Suppose F : J [ n +1] → c . If J is θ -large, where θ = ω α + ω 3 + max { c, || α ||} +3, then there is an α -large set I ⊆ J such that for increasing tuples x, y and x, z in J n +1 , F ( x, y ) = F ( x, z ). where || 0 || = 0 and if α = ω α 1 m 1 + . . . + ω α k m k , then || α || = � k j =1 m j · ( || α j || + 1) Cor (pigeon-hole principle) Let F : J → c . If J is θ -large, where θ = ω α +1 + ω 3 + max { c, || α ||} + 3, then there is an α -large set I ⊆ J on which F is constant. 9
The following simple but wasteful version of Inductive lemma e pigeonhole principle hold Thm (Inductive lemma) Suppose F : J [ n +1] → c , where J is ω k +2 -large and min ( J ) ≥ c . Then there is an ω k -large I ⊆ J such that for in- creasing tuples x, y and x, z in J n +1 , F ( x, y ) = F ( x, z ), or even ( ω k + 1)-large. Cor (pigeon-hole principle) Suppose F : J → c , where J is ω k +2 -large and min ( J ) ≥ c . Then there is an ω k -large I ⊆ J on which F is con- stant. There is also a set that is ( ω k +1)-large. Ramsey Theorem for α -largeness Suppose n ≥ Let F : J [ n ] → c , where J is ω k +2 n -large, 1. consisting of elements ≥ c . Then there is an ω k -large, or even ( ω k + 1)-large homogeneous set I ⊆ J . 10
Thm. Let A be a model of PA . Let Γ be a finite subset of formulas with the free variables split. Let r ∈ N . If J is ( ω 1+2 n ( r − 1) + 1)-large, = x < y → g Γ ( x ) < y , then and for x, y ∈ J , A | there is a subset of J of size r that is diagonal indiscernible for all elements of Γ (n=max length of tuples, g Γ primitive recursive function bounding the number of equivalence classes). 11
Wainer hierarchy: For α < ǫ 0 , F α ( x ) is defined as follows F 0 ( x ) = x + 1, F α +1 ( x ) = F ( x +1) ( x ), α F α ( x ) = max { F { α } ( j ) ( x ) : j ≤ x } for α a limit ordinal Ketonen and Solovay related the notion of α - largeness to the functions of the Wainer Hier- archy. They introduce the function G α ( x ) = µy ([ x, y ] is α − large), Thm. For any α < ǫ 0 (i) F α ( n ) ≤ G ω α ( n + 1); (ii) G ω α ( n ) ≤ F α ( n + 1) Sommer proves Theorem in I ∆ 0 + exp . 12
Refinements of McAloon Thm (McAloon, Som- mer, D’A., Paris, Dimitracopoulos) Let A be a model of I ∆ 0 , and let a be a non- standard element. TFAE: 1) there is an initial segment B of A such that a ∈ B and B is a model of PA ; 2) there is an infinite set I of order type ω , con- sisting of elements greater than a , such that = i 2 < j , and I is diagonal if i < j in I , A | indiscernible for all ϕ ( u, x ) in B 0 ; 3) there exist b and c s.t. c codes satisfaction of bounded formulas by tuples < b , and for all finite r , there is a sequence I r of size r , with = i 2 < j , and I r a < I r < b , s.t. if i < j in I r , A | is diagonal indiscernible for the first r elements of B 0 , 4) there exists b s.t. for all α < ǫ 0 , F α ( a ) ↓ < b . 13
Defn. Let A and B be L -structures. We say that B is an n -elementary substructure of A if for all B n - formulas ϕ ( x ) and all b in B , B | = ϕ ( b ) iff A | = ϕ ( b ). Notation: B ≤ n A , Tarski Criterion: Let B ≤ 0 A , and let n > 0. Suppose that for all B n − 1 formulas ϕ ( x, u ), and for all b in B if there exists d ∈ A such that = ϕ ( b, d ), then there exists d ′ in B such that A | = ϕ ( b, d ′ ). Then B ≤ n A . A | Defn. Let A be a L -structure and let ϕ ( u, x ) be a formula with the free variables splitted into u and x . We say that I bounds witnesses for ϕ ( u, x ) if for all i, j ∈ I such that A | = i < j , and all a ≤ i in A , A | = ( ∃ x ) ϕ ( a, x ) → ( ∃ x ≤ j ) ϕ ( a, x ) . 14
QUESTIONS: 1) Give necessary and sufficient conditions for an initial segment B of A model of I ∆ 0 to be a model of PA and n -elementary substructure of A 2) When does a ∈ A belong to an initial seg- ment B of A model of PA and n -elementary substructure of A ? Lemma: Let A be a model of I ∆ 0 , and let n > 0. Suppose I ⊆ A is a set of order type ω that is diagonal indiscernible for all elements of B 0 and bounds witnesses for all elements of B n − 1 . Let B be the downward closure of I . Then is a model of PA and B ≤ n A . In order to guarantee the existence of such el- ements we distinguish two cases: Case 1: N ≤ n A Case 2: N �≤ n A 15
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