aa r X i v : . [ m a t h . L O ] J u l ON NON-STANDARD MODELS OF ARITHMETICWITH UNCOUNTABLE STANDARD SYSTEMS
WEI WANG
Abstract.
In 1960s, Dana Scott gave a recursion theoretic characterizationof standard systems of countable non-standard models of arithmetic, i.e., col-lections of sets of standard natural numbers coded in non-standard models.Later, Knight and Nadel proved that Scott’s characterization also applies tonon-standard models of arithmetic with cardinality ℵ . But the question,whether the limit on cardinality can be removed from the above characteriza-tion, remains a long standing question, known as the Scott Set Problem. Thisarticle presents two constructions of non-standard models of arithmetic withnon-trivial uncountable standard systems. The first one leads to a new proofof the above theorem of Knight and Nadel, and the second proves the existenceof models with non-trivial standard systems of cardinality the continuum. Apartial answer to the Scott Set Problem under certain set theoretic hypothesisalso follows from the second construction. Introduction
Given a non-standard model of arithmetic M , i.e. a model of arithmetic differentfrom N , a subset of N is coded in M iff it equals to the intersection of N and somedefinable subset of M . The standard system of M , denoted by SSy( M ), is thecollection of subsets of N that are coded in M , and has proved important in thetheory of models of arithmetic. As an example, we recall a theorem of Friedmanand also some related concepts.Given a model M and a finite set of parameters ~a = ( a , . . . , a n ) from M , a type p of M over ~a is recursive , iff p is in a fixed finite set of free variables ~x and thefollowing set of formulas is recursive { ϕ ( ~x, ~y ) : ϕ ( ~x, ~a ) ∈ p } . A model M is recursively saturated iff every recursive type of M is realized in M .Every infinite model can be elementarily extended to a recursively saturated model.For countable recursively saturated models of PA, standard systems are their ownblueprints kept by themselves, according to Friedman’s Embedding Theorem below(which can also be found in more recent literature like [6]). Theorem 1.1 (Friedman [2]) . Suppose that M and N are countable recursivelysaturated models of PA , and they are elementarily equivalent. Then SSy( M ) ⊆ SSy( N ) iff there exists an elementary embedding sending M to an initial segmentof N . Moreover, M and N are isomorphic iff SSy( M ) = SSy( N ) . In 1962, Scott [7] proved that the standard system S of a non-standard model(of arithmetic) always satisfies some recursion theoretic conditions below. Mathematics Subject Classification.
Key words and phrases.
Non-standard models of arithmetic, standard systems, Scott sets,Martin’s Axiom.The author was partially supported by China NSF Grant 11971501. The results here have beenpresented in several occasions. The author thanks various logicians for their helpful opinions, inparticular, Tin Lok Wong, Jiacheng Yuan, Yinhe Peng and Victoria Gitman. (S1) If X and Y are both in S then so is X ⊕ Y = 2 X ∪ (2 Y + 1).(S2) If X ∈ S and Y is recursive in X then Y ∈ S .(S3) If S contains an infinite binary tree T then S also contains an infinite pathof T .Today, a collection of subsets of N satisfying (S1-3) above is called a Scott set . Scottalso proved the reverse direction for countable Scott sets.
Theorem 1.2 (Scott [7]) . A countable S is a Scott set iff S = SSy( M ) for somecountable non-standard model M of PA . Knight and Nadel [5] extended Scott’s Theorem to some uncountable Scott sets.
Theorem 1.3 (Knight and Nadel [5]) . Every Scott set of cardinality ≤ ℵ is thestandard system of a non-standard model of PA . But the question, whether Scott’s Theorem holds for arbitrary Scott sets, remainsopen, and has been named the
Scott Set Problem in literature (e.g., see [6]).
Question 1.4 (Scott Set Problem) . Does every Scott set equal to
SSy( M ) of somenon-standard model of PA ? Nevertheless, there are some interesting partial answers to the Scott Set problem.For example, Gitman [3] proved that certain uncountable Scott sets could equal tostandard systems of non-standard models, under the Proper Forcing Axiom. Peoplealso investigate parallel questions in other first order theories, e.g., real closed fieldsand Presburger arithmetic in [1].This article presents some attempts to understand the Scott Set Problem.In §
2, we shall see an alternative proof of Theorem 1.3 of Knight and Nadel.Indeed, there have been several alternative proofs of Theorem 1.3. It may be inter-esting to note that all known proofs of Theorem 1.3 rely on recursively saturatedmodels, including the original proof and a recent one published in [1]. These may beread as evidences reinforcing the tie between standard systems and recursively satu-rated models. However, the proof presented here does not need recursive saturationand looks more straightforward.In §
3, we shall prove that there do exist models of PA which have non-trivialstandard systems of cardinality the continuum. From the proof of this existence,we shall be able to derive some partial answer to the Scott Set Problem.We finish this section by recalling some notations and basic knowledge whichwill be used in the rest of the article.Above we use N to denote the standard model of arithmetic. We shall alsouse Q to denote the set of standard rational numbers. But in many cases, it ismore convenient to use ω for N , as in set theory. A subset of ω is identified withits characteristic function. Given a non-standard M | = PA, every a ∈ M can beregarded as a binary sequence, with the i -th bit denoted by ( a ) i . If a ∈ M − ω , a codes the following subset of ω { i ∈ ω : M | = ( a ) i = 1 } . The standard system of M , denoted by SSy( M ), is the collection of subsets of ω coded by some a ∈ M − ω . It is easy to see that the elements of SSy( M ) coincidewith subsets of ω which are intersections of ω and definable subsets of M .Since PA admits definable Skolem functions, we can build an elementary exten-sion of a given M | = PA, by building a type p ( x ) of M , and then take an extension(called a p ( x ) -extension of M ) with its universe consisting of F ( b ), where b is afixed realization of p ( x ) and F ranges over all unary functions definable in M . If p ( x ) is bounded, i.e., p ( x ) ⊢ x < a for some a ∈ M , we may even assume that F isdefined on { i ∈ M : i < a } and so is (coded by) an element of M . N NON-STANDARD MODELS OF ARITHMETIC 3
A collection S of subsets of ω satisfying (S1,S2) above is called a Turing ideal .So Scott sets are Turing ideals satisfying (S3). Suppose that I is a Turing ideal.A set is I -recursive iff it is recursive in some set in I . Given another X ⊆ ω , let I ⊕ X denote the following collection { Z ⊆ ω : Z is recursive in I ⊕ X } , which is clearly also a Turing ideal.For a better background in models of arithmetic, we recommend [6].2. A Straightforward Construction
This sections presents an alternative proof of Theorem 1.3 of Knight and Nadel,via the following result of Ehrenfeucht. It is clear that Theorem 2.1 implies Theorem1.3. Theorem 2.1 itself is known provable via Friedman’s Embedding Theorem 1.1(see [3]). So the known proof relies on recursive saturation. Below we presenta proof of Theorem 2.1 via a straightforward construction, which does not needrecursive saturation.
Theorem 2.1 (Ehrenfeucht) . Let S be a Scott set and M a countable non-standardmodel of PA with SSy( M ) ⊆ S . For every X ∈ S there exists a countable elementaryextension N of M with X ∈ SSy( N ) ⊆ S . Let S , M and X be as in the statement of the above theorem. Fix a ∈ M − ω .We shall construct a type p ( x ) of M s.t. p ⊢ x < a and then let N be a p ( x )-extension of M . As M is countable, N will be countable as well. The type p ( x )will be constructed as a union of types ( p i ( x ) : i ∈ ω ).As M is countable, we can fix a list ( f i : i ∈ ω ) of all f ∈ M which maps2 a = { n ∈ M : n < a } to M . Assume that f is the identify function on 2 a .Let p ( x ) = { x < a } ∪ { ( x ) n = X ( n ) : n ∈ ω } = { x < a } ∪ { ( f ( x )) n = X ( n ) : n ∈ ω } . As a > ω , p ( x ) is finitely realizable in M . Also note that p ( x ) is recursive in X ,and if N is a p ( x )-extension of M then X ∈ SSy( N ).Suppose that for k ∈ ω we have the following data • X , . . . , X k ⊆ ω s.t. X = X and each X i is in S ; • A type of M as below p k ( x ) = { x < a } ∪ { ( f i ( x )) n = X i ( n ) : i ≤ k, n ∈ ω } . Note that p k ( x ) is recursive in L i ≤ k X and thus recursive in S , and that if N is a p k ( x )-extension of M and b ∈ N realizes p k ( x ) then f i ( b ) codes X i for all i ≤ k .Let T be the set of tuples ~σ = ( σ i : i ≤ k + 1) s.t. σ i ’s are finite binary sequencesof equal length and in M the following set is not empty W ( ~σ ) = { c < a : ∀ i ≤ k + 1 , n < | σ i | (( f i ( c )) n = σ i ( n )) } . So T is in SSy( M ).Fix m ∈ ω . For each i ≤ k , let σ i be the initial segment of X i of length m . As p k ( x ) is finitely realizable in M , there exists c ∈ M s.t. c < a and ( f i ( c )) n = σ i ( n )for each i ≤ k and n < m . Define a binary sequence σ k +1 of length m by letting σ k +1 ( n ) = ( f i ( c )) n for n < m . Then for this tuple ~σ = ( σ i : i ≤ k + 1), the set W ( ~σ ) contains c and thus is not empty. So ~σ ∈ T . This shows that T is infinite.Let T ′ be the set of τ ∈ <ω s.t. if τ i is the initial segment of X i of length | τ | then ( τ , . . . , τ k , τ ) ∈ T . By the above paragraph, T ′ is an infinite binary treerecursive in L i ≤ k X i ⊕ T and thus in S . So by (S3) in the definition of Scott set, S contains an infinite path of T ′ , denoted by X k +1 . WEI WANG
Hence the following set is a type of M , p k +1 ( x ) = p k ( x ) ∪ { ( f k +1 ( x )) n = X k +1 ( n ) : ∀ n ∈ ω } , and p k +1 ( x ) is recursive in S .Finally, let p ( x ) = S k p k ( x ). Then p ( x ) is a type of M , and if b realizes p ( x )then b codes X and each f i ( b ) codes X i which is in S . So any p ( x )-extension of M is a desired model N .This ends the proof of Ehrenfeucht’s Theorem 2.1.3. Uncountable Standard Systems
Here we shall prove the existence of non-standard models whose standard systemsare non-trivial and have cardinality the continuum.
Theorem 3.1 (ZF) . For every non-standard countable N | = PA , there are ( M X : X ⊆ ω ) s.t. each M X is an elementary extension of N , | M X | = | SSy( M X ) | =max { ω, |X |} and X ⊆ Y ⇔ M X (cid:22) M Y ⇔ SSy( M X ) ⊆ SSy( M Y ) . Moreover, if
A ⊂ ω − SSy( N ) is countable then we can have A ∩
SSy( M X ) = ∅ for all X ⊆ ω . Fix a ∈ N − ω . For types, we shall mean types of N .For each n ≤ ω and σ ∈ n , let x σ be a variable. If m ≤ n , σ , . . . , σ k ∈ n and φ ( x σ , . . . , x σ k ) contains no quantifiers over any x σ i , then the m -reduct of φ is theformula φ ( x σ , . . . , x σ k ; x σ ↾ m , . . . , x σ k ↾ m ) , i.e., the formula obtained by simultaneously substituting x σ i ↾ m ’s for x σ i ’s in φ ,where σ ↾ m is the sequence consisting of the first m bits of σ . We also call theoriginal φ an n -ramification of its m -reduct.A condition p is a finite type in ( x σ : σ ∈ n p ) for some n p ∈ ω , s.t. p containsno quantifiers over any x σ , p ⊢ x σ ∈ a and there exists a positive r ∈ Q with N | = | p ( N ) | > r (2 a ) np , where p ( N ) is the set of realizations of p in N . Let P be the set of conditions. For p, q ∈ P , q ≤ p iff n q ≥ n p and q contains every n q -ramification of every φ ∈ p .For a descending sequence ~p = ( p i : i ∈ ω ) from P s.t. lim i n p i = ∞ , let G ~p bethe set of φ ( x f , . . . x f k ) ( f i ∈ ω ) s.t. the n p i -reduct of φ is in p i for some i . Lemma 3.2. If ~p and G ~p are as above then G ~p is a type of N .Proof. For every finite subset H of G ~p , there is a fixed i s.t. formulas in H areramifications of formulas in p i . As p i is a finite type of N , p i is realized in N bysome tuple, which also realizes H . (cid:3) To construct G ~p as above, we should be able to extend conditions non-trivially. Lemma 3.3.
Each condition p can be extended to another condition q with n q > n p .Proof. Let q be the set of ( n p + 1)-ramifications of all formulas in p . Then q is asdesired. (cid:3) The lemma below will be used to that if X and Y are different subsets of 2 ω thenSSy( M X ) and SSy( M Y ) are different. Lemma 3.4.
Suppose that p ∈ P and F : N k → N is definable in N . Then thereexists q ≤ p s.t. n q = n p and every ( σ, σ , . . . , σ k ) from n q with σ = σ , . . . , σ k corresponds to some i < ω with q ⊢ ( x σ ) i = ( F ( x σ , . . . , x σ k )) i . N NON-STANDARD MODELS OF ARITHMETIC 5
Proof.
It suffices to prove that every ( σ, σ , . . . , σ k ) from 2 n p with σ = σ , . . . , σ k corresponds to some i < ω and q ≤ p s.t. n q = n p and q ⊢ ( x σ ) i = ( F ( x σ , . . . , x σ k )) i .Let n = n p , r ∈ Q be positive s.t. | p ( N ) | > r (2 a ) n in N . Fix ( σ, σ , . . . , σ k )from 2 n as above. For each m ∈ ω , in N the cardinality of the following set { ( b τ : τ ∈ n ) ∈ p ( N ) : ∀ i < m (( b σ ) i = ( F ( b σ , . . . , b σ k )) i ) } is at most 2 − m (2 a ) n , since each ( b τ : τ ∈ n ) in the set has the first m bits of b σ determined by ( b σ , . . . , b σ k ). Hence, there must be some i ∈ ω and some positive ǫ ∈ Q s.t. in N , |{ ( b τ : τ ∈ n ) ∈ p ( N ) : ( b σ ) i = ( F ( b σ , . . . , b σ k )) i }| > ǫ (2 a ) n . So q = p ∪ { ( x σ ) i = ( F ( x σ , . . . , x σ k )) i } is as desired. (cid:3) To exclude certain g ’s from SSy( M X )’s, we prove one more lemma below. Lemma 3.5. If g SSy( N ) , F : N k → N is definable in N and p ∈ P then thereexists q ≤ p s.t. n q = n p and every ( σ , . . . , σ k ) ∈ (2 n q ) k corresponds to some i < ω with q ⊢ g ( i ) = ( F ( x σ , . . . , x σ k )) i .Proof. It suffices to prove that every ( σ , . . . , σ k ) ∈ (2 n q ) k corresponds to some q ≤ p and i < ω s.t. n q = n p and q ⊢ g ( i ) = ( F ( x σ , . . . , x σ k )) i .Fix all the data and ( σ , . . . , σ k ) ∈ (2 n q ) k as above. Define a function h : ω → h ( i ) be the least j <
2, s.t. in N , |{ ( b τ : τ ∈ n p ) ∈ p ( N ) : j = ( F ( b σ , . . . , b σ k )) i }| ≥ | p ( N ) | / . So h ∈ SSy( N ). Since g SSy( N ), we can pick i < ω s.t. g ( i ) = h ( i ). Then q = p ∪ { g ( i ) = ( F ( x σ , . . . , x σ k )) i } is as desired. (cid:3) By the above lemmata, we can construct ~p = ( p i : i ∈ ω ) s.t.(1) p i +1 ≤ p i ∈ P ;(2) lim i n p i = ∞ ;(3) For each p i and each N -definable function F : N k → N , there exist p j ≤ p i and m < ω , s.t. if σ , . . . , σ k ∈ n pj and σ ∈ n pj − { σ , . . . , σ k } then p j ⊢ ∃ n < m (( x σ ) n = ( F ( x σ , . . . , x σ k )) n );(4) For each p i , each g ∈ A and each N -definable function F : N k → N , thereexist p j ≤ p i and m < ω s.t. every σ , . . . , σ k ∈ (2 n pj ) k corresponds tosome n < m with p j ⊢ g ( n ) = ( F ( x σ , . . . , x σ k )) n .So G ~p is a type in ( x f : f ∈ ω ) over N . Let ( a f : f ∈ ω ) be a realization of G ~p insome N ′ ≻ N . If X ⊆ ω , let M X be the Skolem hull of N ∪ { a f : f ∈ X } in N ′ .Then M X ’s ( X ⊆ ω ) are as desired.This finishes the proof of Theorem 3.1. Corollary 3.6 (ZFC + MA) . For every non-standard N | = PA s.t. | N | < ω ,there exists a family ( M X : X ⊆ ω ) s.t. N ≺ M X , | M X | = | SSy( M X ) | =max {| SSy( N ) | , |X |} and X ⊆ Y ⇔ M X (cid:22) M Y ⇔ SSy( M X ) ⊆ SSy( M Y ) . Moreover, if
A ⊂ ω − SSy( N ) has cardinality < ω then we can have A ∩
SSy( M X ) = ∅ for all X ⊆ ω .Proof. It is easy to see that the poset P in the proof of Theorem 3.1 satisfies thecountable chain condition, even if N is uncountable. By ZFC + MA, we can applyLemmata 3.3, 3.4 and 3.5 to N and A both of cardinality less than the continuum,and obtain a filter F ⊂ P s.t.(1) Each p ∈ F has an extension q ∈ F with n q > n p ; WEI WANG (2) For each p ∈ F and each N -definable function F : N k → N , there exist q ∈F and m < ω , s.t. q ≤ p , and if σ , . . . , σ k ∈ n q and σ ∈ n q − { σ , . . . , σ k } then q ⊢ ∃ n < m (( x σ ) n = ( F ( x σ , . . . , x σ k )) n );(3) For each p ∈ F , each g ∈ A and each N -definable function F : N k → N , there exist q ∈ F and m < ω s.t. q ≤ p , every σ , . . . , σ k ∈ (2 n q ) k corresponds to some n < m with q ⊢ g ( n ) = ( F ( x σ , . . . , x σ k )) n .Then we define G F to be the set of formulas in ( x f : f ∈ ω ) s.t. every formula in G F has a reduct in some p ∈ F . It can be proved that G F is a type of N , similarto Lemma 3.2. Finally, take a realization ( a f : f ∈ ω ) of G F and let M X be anextension of N generated by N ∪ { a f : f ∈ X } . (cid:3) Corollary 3.6 can be extended to a partial answer to the Scott Set Problem.
Corollary 3.7 (ZFC + MA) . Suppose that M is a countable non-standard modelof PA , A ⊂ ω is of cardinality < ω , and B ⊂ ω is countable and s.t. the Turingideal generated by SSy( M ) ∪ B is disjoint from A . Then there exists a family ( M X : X ⊆ ω ) s.t. M ≺ M X , B ⊆
SSy( M X ) , | M X | = | SSy( M X ) | = max { ω, |X |} ,and X ⊆ Y ⇔ M X (cid:22) M Y ⇔ SSy( M X ) ⊆ SSy( M Y ) . Proof.
By MA and well-known recursion theoretic technique (e.g., see [4, Lemma2.6]), we can construct a countable Scott set S s.t. SSy( M ) ∪ B ⊆ S and A ∩ S = ∅ .By Ehrenfeucht’s Theorem 2.1, M has an elementary extension N with SSy( N ) = S . The conclusion then follows from an application of Corollary 3.6 to N and A . (cid:3) By Corollary 3.7, for a Scott set S which is possibly of cardinality the continuum,if we pick A ⊆ ω −S of cardinality less than the continuum and also a countable B ⊆S , then we can find a non-standard M | = PA s.t. | SSy( M ) | = 2 ω , A ∩
SSy( M ) = ∅ and B ⊂
SSy( M ). So Corollary 3.7 can be regarded as a partial answer to the ScottSet Problem. References [1] Alf Dolich, Julia F. Knight, Karen Lange, and David Marker. Representing Scott sets inalgebraic settings.
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J. Symbolic Logic , 73(3):845–860, 2008.[4] Carl G. Jockusch, Jr. and Robert I. Soare. Π classes and degrees of theories. Trans. Amer.Math. Soc. , 173:33–56, 1972.[5] Julia Knight and Mark Nadel. Models of arithmetic and closed ideals.
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Institute of Logic and Cognition and Department of Philosophy, Sun Yat-Sen Uni-versity, Guangzhou, China
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