Search

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ4), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

Bounded lattices (that is lattices that are both lower bounded and upper bounded) form a large class of lattices that include all distributive lattices, many nondistributive finite lattices such as the pentagon lattice N5. and all lattices in any variety generated by a finite bounded lattice. Extending a theorem of Paris for distributive lattices, we prove that if L is an ℵ0-algebraic bounded lattice, then every countable nonstandard model of Peano Arithmetic has a cofinal elementary extension such that the interstructure lattice Lt(/) is isomorphic to L.

A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model , i.e., the elementary diagram De () has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree.

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, IΔ1. We show that IΔ1 is independent from the set of all true arithmetical Π2-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of Δ1-induction.An open problem formulated by J. Paris (see [4, 5]) is whether IΔ1 proves the corresponding least element principle for decidable predicates, LΔ1 (or, equivalently, the Σ1-collection principle BΣ1). We reduce this question to a purely computation-theoretic one.

Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13], [18]. We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π1 sentence, valid in the standard model of the Natural Numbers and denoted as V. such that if α is any finite consistent extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property.Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ0, as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].)

We generalize a result on True Arithmetic (ℐA) by Lachlan and Soare to certain other completions of Peano Arithmetic (PA). If ℐ is a completion of PA, then Rep(ℐ) denotes the family of sets X ⊆ ω for which there exists a formula φ(x) such that for all n ∈ ω, if n ∈ X, then ℐ ⊢ φ(S(n) (0)) and if n ∉ X, then ℐ ⊢ ┐φ(S(n)(O)). We show that if S, J ⊆ P(ω) such that S is a Scott set, J is a jump ideal, S ⊂ J and for all X ∈ J, there exists C ∈ S such that C is a “coding” set for the family of subtrees of 2 computable in X, and if ℐ is a completion of PA Such that Rep(ℐ) = S, then there exists a model A of ℐ such that J is the Scott set of A and no enumeration of Rep(ℐ) is computable in A. The model A of ℐ is obtained via a new notion of forcing.Before proving our main result, we demonstrate the existence of uncountably many different pairs (S, J) satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2. In particular, C C ⊆ ω is a coding set for the family of subtrees of 2 computable in X if and only if for all trees T ⊆ 2 computable in X, if χc is a path through T, then there exists σ ∈ T such that σ ⊂ χc and every extension of σ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2. We show they are identical.

We prove that a necessary and sufficient condition for a countable set of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: is closed under arithmetical definability and contains 0(ω) the set of all (Gödel numbers of) true arithmetical sentences.Some results related to definability of sets of integers in elementary extensions of ω are included.

A new method of “minimal” readability is proposed and applied to show that the definable functions of Heyting arithmetic (HA)—functions f such that HA ⊢ ∀x∃!yA(x, y) ⇒ for all m, A(m, f(m)) is true, where A(x, y) may be an arbitrary formula of ℒ(HA) with only x,y free—are precisely the provably recursive functions of the classical Peano arithmetic (PA), i.e., the < ε0-recursive functions. It is proved that, for prenex sentences provable in HA, Skolem functions may always be chosen to be < ε0-recursive. The method is extended to intuitionistic finite-type arithmetic, , and elementary analysis. Generalized forms of Kreisel's characterization of the provably recursive functions of PA and of the no-counterexample-interpretation for PA are consequently derived.

We prove that in IST, Nelson's internal set theory, the Uniqueness and Collection principles, hold for all (including external) formulas. A corollary of the Collection theorem shows that in IST there are no definable mappings of a set X onto a set Y of greater (not equal) cardinality unless both sets are finite and #(Y) ≤ n #(X) for some standard n. Proofs are based on a rather general technique which may be applied to other nonstandard structures. In particular we prove that in a nonstandard model of PA, Peano arithmetic, every hyperinteger uniquely definable by a formula of the PA language extended by the predicate of standardness, can be defined also by a pure PA formula.

In this paper the modal operator “x is provable in Peano Arithmetic” is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.

The aim of this paper is to extend the Bass-Milnor-Serre theorem to the nonstandard rings associated with nonstandard models of Peano arithmetic, in brief to Peano rings.First, we recall the classical setting. Let k be an algebraïc number field, and let θ be its ring of integers. Let n be an integer ≥ 3, and let G be the group Sln(θ) of (n, n) matrices of determinant 1 with coefficients in θ.The profinite topology in G is the topology having as fundamental system of open subgroups the subgroups of finite index.Congruence subgroups of finite index of G are the kernels of the maps Sln(θ) → Sln(θ/I) for which all ideals I of θ are of finite index. By taking these subgroups as a fundamental system of open subgroups, one obtains the congruence topology on G. Every open set for this topology is open in the profinite topology.We denote by Ḡ (resp., Ĝ) the completion of G for the congruence (resp., profinite) topology.The Bass-Milnor-Serre theorem [1] consists of the two following statements:(A) If k admits a real embedding, then we have an exact sequenceThat is, Ĝ and Ḡ are isomorphic.(B) If k is totally imaginary, then one has an exact sequencewhere μ(k)is the group of the roots of unity of k.

Kruskal proved that finite trees are well-quasi-ordered by hom(e)omorphic embeddability. Friedman observed that this statement is not provable in predicative analysis. Friedman also proposed (see in [Simpson]) some stronger variants of the Kruskal theorem dealing with finite labeled trees under hom(e)omorphic embeddability with a certain gap-condition, where labels are arbitrary finite ordinals from a fixed initial segment of ω. The corresponding limit statement, expressing that for all initial segments of ω these labeled trees are well-quasi-ordered, is provable in -CA, but not in the analogous theory -CA0 with induction restricted to sets. Schütte and Simpson proved that the one-dimensional case of Friedman's limit statement dealing with finite labeled intervals is not provable in Peano arithmetic. However, Friedman's gap-condition fails for finite trees labeled with transfinite ordinals. In [Gordeev 1] I proposed another gap-condition and proved the resulting one-dimensional modified statements for all (countable) transfinite ordinal-labels. The corresponding universal modified one-dimensional statement UM1 is provable in (in fact, is equivalent to) the familiar theory ATR0 whose proof-theoretic ordinal is Γ0. In [Gordeev 1] I also announced that, in the general case of arbitrarily-branching finite trees labeled with transfinite ordinals, in the proof-theoretic sense the hierarchy of the limit modified statements M (which are denoted by LMλ in the present note) is as strong as the hierarchy of the familiar theories of iterated inductive definitions (more precisely, see [Gordeev 1, Concluding Remark 3]). In this note I present a “positive” proof of the full universal modified statement UM, together with a short proof of the crucial “reverse” results which is based on Okada's interpretation of the well-established ordinal notations of Buchholz corresponding to the theories of iterated inductive definitions. Formally the results are summarized in §5 below.

We present a theory VF of partial truth over Peano arithmetic and we prove that VF and ID1, have the same arithmetical content. The semantics of VF is inspired by van Fraassen's notion of supervaluation.

In this paper we show how to interpret Robinson’s Arithmetic Q and the theory R of Tarski, Mostowski and Robinson as theories of cardinals in very weak theories of relations over a domain.

There is an analogy between concepts such as end-extension types and minimal types in the model theory of Peano arithmetic and concepts such as P-points and selective ultrafilters in the theory of ultrafilters on N. Using the notion of conservative extensions of models, we prove some theorems clarifying the relation between these pairs of analogous concepts. We also use the analogy to obtain some modeltheoretic results with techniques originally used in ultrafilter theory. These results assert that every countable nonstandard model of arithmetic has a bounded minimal extension and that some types in arithmetic are not 2-isolated.. Introduction. In this paper we shall analyze (in ?2) and exploit (in ??3, 4) a certain analogy between some concepts in the theory of models of Peano arithmetic and some concepts in the theory of ultrafilters on the set N of natural numbers. As might be expected, we shall make considerable use of the fact that an ultrafilter on N may be viewed as the type realized by an element in a suitable nonstandard model of arithmetic, provided that our "arithmetic" contains symbols for all relations on N. (But we shall not use the related fact that a type in Peano arithmetic may be viewed as an ultrafilter in a suitable Lindenbaum algebra.) The difference between the uncountable language with symbols for all relations and the countable language of Peano arithmetic affects the spirit of proofs in the two theories. Constructions of ultrafilters with interesting properties are usually transfinite inductions of length the power of the continuum. Difficulties often occur at limit stages of the induction, and many theorems depend on the continuum hypothesis (or some related assumptions, like Martin's axiom) to make the limit stages tractable. On the other hand, constructions of models or types in Peano arithmetic are usually ordinary inductions (of length co), so there are no limit stages to worry about, but the successor stages require more work than in ultrafilter theory because one has to know that sets with certain properties not only exist but are definable in the language under consideration (and provably have the properties one wants). These features of typical proofs in the two theories can be clearly seen in Rudin's proof [12] that the continuum hypothesis implies the existence of Ppoints and MacDowell and Specker's proof [9] that every model of Peano arithmetic has an elementary end-extension. If one ignores the differences indicated above between these two proofs, one finds that they both depend on the same simple fact: If a function f has an infinite domain, then f is either constant or one-to-one on some infinite subset of the domain. The connection between the two existence proofs may be taken as evidence of some connection between P-points and end-extensions (or, better, end-extension Received July 2, 1973. (D 1974, Association for Symbolic Logic

Let PA be Peano arithmetic with function symbols′, + and ·. The length of a proof P, denoted by lh(P), is the maximum length of threads of P (for the term ‘thread’, see [T, p. 14]). For a formula A and a natural number m, PA ⊢mA denotes the fact that there is a proof in PA of A whose length is less than or equal to m. PA ⊢ A denotes the fact that there is a proof in PA of A.G. Kreisel conjectured that the following proposition holds.“Let m be a natural number and A(a) be a formula. If for each natural number n, then PA ⊢ ∀xA(x)”.Let PA1 be the corresponding system with + and · replaced by ternary predicates and axioms saying that these predicates represent functions. Parikh [P] proved the following proposition which is obtained from Kreisel's conjecture by replacing PA by PA1.Proposition. Let A(a) be a formula in PA1and m be a natural number. Assume thatfor each natural number n. Then PA1 ⊢ ∀xA(x).The reason why Parikh's method succeeds is the fact that the only function symbol ′ in PA1 is unary. So his method fails for PA.To solve this conjecture for PA, we must make syntactical investigation into proofs in PA of formulas of the form A() with length ≤ m. Even if lengths of proofs are less than or equal to m, depths of occurrences of bound variables in induction axiom schemata or equality axiom schemata in proofs are not always bounded.

This paper investigates the quantifier “there exist unboundedly many” in the context of first-order arithmetic. An alternative axiomatization is found for Peano arithmetic based on an axiom schema of regularity: The union of boundedly many bounded sets is bounded. We also obtain combinatorial equivalents of certain second-order theories associated with cuts in nonstandard models of arithmetic.

We give two new finite combinatorial statements which are independent of Peano arithmetic, using the methods of Kirby and Paris [6] and Paris [12]. Both are in fact equivalent over Peano arithmetic (denoted by P) to its 1-consistency. The first involves trees and the second linear orderings. Both were “motivated” by anti-basis theorems of Clote (cf. [1], [2]). The one involving trees, however, is not unrelated to the Kirby-Paris characterization of strong cuts in terms of the tree property [6], but, in fact, comes directly from König's lemma, of which it is a miniaturization. (See the remark preceding Theorem 3 below.) The resulting combinatorial statement is easily seen to imply the independent statement discovered by Mills [11], but it is not clear how to show their equivalence over Peano arithmetic without going through 1-consistency. The one involving linear orderings miniaturizes the property of infinite sets X that any linear ordering of X is isomorphic to ω or ω* on some infinite subset of X. Both statements are analogous to Example 2 of [12] and involve the notion of dense [12] or relatively large [14] finite set.We adopt the notations and definitions of [6] and [12]. We shall in particular have need of the notions of semiregular, regular and strong initial segments and of indicators.

This paper is a sequel to our earlier paper [2] entitled Saturation and simple extensions of models of Peano arithmetic. Among other things, we will answer some of the questions that were left open there. In §1 we consider the question of whether there are lofty models of PA which have no recursively saturated, simple extensions. We are still unable to answer this question; but we do show in that section that these models are precisely the lofty models which are not recursively saturated and which are κ-like for some regular κ. In §2 we use diagonal methods to produce minimal models of PA in which the standard cut is recursively definable, and other minimal models in which the standard cut is not recursively definable. In §3 we answer two questions from [2] by exhibiting countable models of PA which, in the terminology of this paper, are uniformly ω-lofty but not continuously ω-lofty and others which are continuously ω-lofty but not recursively saturated. We also construct a model (assuming ◇) which is not recursively saturated but every proper, simple cofinal extension of which is ℵ1-saturated. Finally, in §4 we answer another question from [2] by proving that for regular κ ≥ ℵ1; every κ-saturated model of PA has a κ-saturated proper, simple extension which is not κ+-saturated.Our notation and terminology are quite standard. Anything unfamiliar to the reader and not adequately denned here is probably defined in §1 of [2]. All models considered are models of Peano arithmetic.

We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N(x) for an elementary initial segment, along with axiom schemes approximating ω1-saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.

This paper is devoted to the study of recursively and short recursively saturated models of PA by means of so-called nonstandard satisfaction methods. The paper is intended to be self-contained. In particular, no knowledge of nonstandard satisfaction classes is assumed. In fact we shall not use this notion explicitly.We define a certain property of models of PA which we call the S-property and prove that properly short recursively saturated models (see Definition 2.1. below) are exactly short models with the S-property. The main result is that all properly short recursively saturated models are elementary cuts of recursively saturated models. This is a generalization to the uncountable case of the theorem of C. Smorynski [9] and is an easy application of some general results concerning cofinal extensions of models of PA which we discuss in §3.On the way we obtain another proof of the result of Smorynski and Stavi [10] which says that short recursive and recursive saturation is preserved under cofinal extensions.The author wants to thank H. Kotlarski and W. Marek for valuable suggestions concerning the subject of the paper.Special thanks must also go to J. Paris for the lemma used in the proof of Theorem 3.5.

In this paper we shall outline a purely model theoretic method for obtaining independence results for Peano's first order axioms (P). The method is of interest in that it provides for the first time elementary combinatorial statements about the natural numbers which are not provable in P. We give several examples of such statements.Central to this exposition will be the notion of an indicator. Indicators were introduced by L. Kirby and the author in [3] although they had occurred implicitly in earlier papers, for example Friedman [1]. The main result on indicators which we shall need (Lemma 1) was proved by Laurie Kirby and the author in the summer of 1976 but it was not until early in the following year that the author realised that this lemma could be used to give independence results.The first combinatorial independence results obtained were essentially statements about certain finite games and consequently were not immediately meaningful (see Example 2). This shortcoming was remedied by Leo Harrington who, upon hearing an incorrect version of our results, noticed a beautifully simply independent combinatorial statement. We outline this result in Example 3. An alternative, more detailed, proof may be found in [5].Clearly Laurie Kirby and Leo Harrington have made a very significant contribution to this paper and we wish to express our sincere thanks to them.

We construct a model of Peano arithmetic in an uncountable language which has no elementary end extension. This answers a question of Gaifman and contrasts with the well-known theorem of MacDowell and Specker which states that every model of Peano arithmetic in a countable language has an elementary end extension. The construction employs forcing in a nonstandard model.

For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size 2, with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting types over T, H(T), is at least 2, (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly 2b,.) If TX true arithmetic, then H(T) = 2,. If 8 -A (p)'-, then any completion of Peano Arithmetic has a model of size 8 with no set of indiscernibles of size p. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = L. ?0. Introduction. The main accomplishment of the research presented here is the lifting of the Specker-MacDowell-Gaifman technology for 1-types over arithmetic to a technique for building models from carefully constructed n-types. In Specker and MacDowell [1959] it is shown that every model of Peano Arithmetic, (PA), has a proper elementary end extension. Gaifman [1965], [1968] shows how to construct "end extension 1-types" which give rise to such extensions, investigates the iteration of such extension techniques, and gives a construction of "mutually independent end extension types" which is the direct forebear of the "mutually generic types" we use here. It is implicit in these papers (and, in fact, is a special case of the M(I) in Gaifman [1976]) that a model of PA generated by 21-many mutually generic types will have no two distinct elements with the same 1-type.3 Here we give a direct exposition of such a model, in a way that clearly suggests how to construct a model of PA with no (n + 1)-length sequence of indiscernibles. Carrying through this suggestion involves far more complicated combinatorics than in the case of 2i. What is needed is a certain generalization of finitary Ramsey's Theorem proved independently by Neset'ril and Rodl [1976] and by the present authors. Our original exposition also involved some complicated machinery specially designed for our particular results. We owe a large debt to Haim Gaifman who pointed out some structural properties of our models which are of interest in their own right and which can be used directly to obtain our results. The approach of the present paper is that suggested to us by Gaifman, and we will clearly indicate the specific results which are due to him. Received April 19, 1976; revised September 12, 1977. University of Wisconsin-Milwaukee. This author's research was supported, in part, by an American Mathematical Society Postdoctoral Research Fellowship and by NSF Grant Number