A standard model of Peano arithmetic with no conservative elementary extension

AbstractWe 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.

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Two theorems on degrees of models of true arithmetic

Journal of Symbolic Logic ◽

10.2307/2274174 ◽

1984 ◽

Vol 49 (2) ◽

pp. 425-436 ◽

Cited By ~ 7

Author(s):

Julia Knight ◽

Alistair H. Lachlan ◽

Robert I. Soare

Keyword(s):

Standard Model ◽

Binary Operation ◽

Peano Arithmetic ◽

Turing Degree ◽

First Order ◽

Nonstandard Model ◽

Basis Theorem ◽

The Standard Model ◽

The Universe ◽

Countable Models

Let PA be the theory of first order Peano arithmetic, in the language L with binary operation symbols + and ·. Let N be the theory of the standard model of PA. We consider countable models M of PA such that the universe ∣M∣ is ω. The degree of such a model M, denoted by deg(M), is the (Turing) degree of the atomic diagram of M. The results of this paper concern the degrees of models of N, but here in the Introduction, we shall give a brief survey of results about degrees of models of PA.Let D0 denote the set of degrees d such that there is a nonstandard model of M of PA with deg(M) = d. Here are some of the more easily stated results about D0.(1) There is no recursive nonstandard model of PA; i.e., 0 ∈ D0.This is a result of Tennenbaum [T].(2) There existsd ∈ D0such thatd ≤ 0′.This follows from the standard Henkin argument.(3) There existsd ∈ D0such thatd < 0′.Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem.(4) There existsd ∈ D0such thatd′ = 0′.Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4).(5) D0 = Dc = De, where Dc denotes the set of degrees of completions of PA and De the set of degrees d such that d separates a pair of effectively inseparable r.e. sets.Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that Dc is upward closed.

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Infinite substructure lattices of models of Peano Arithmetic

Journal of Symbolic Logic ◽

10.2178/jsl/1286198152 ◽

2010 ◽

Vol 75 (4) ◽

pp. 1366-1382

Author(s):

James H. Schmerl

Keyword(s):

Large Class ◽

Peano Arithmetic ◽

Distributive Lattices ◽

Elementary Extension ◽

Bounded Lattice ◽

Nonstandard Model ◽

Finite Lattices ◽

Bounded Form

AbstractBounded 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.

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Flipping properties in arithmetic

Journal of Symbolic Logic ◽

10.2307/2273151 ◽

1982 ◽

Vol 47 (2) ◽

pp. 416-422 ◽

Cited By ~ 3

Author(s):

L. A. S. Kirby

Keyword(s):

Standard Model ◽

Set Theory ◽

Initial Segment ◽

Peano Arithmetic ◽

Combinatorial Properties ◽

Nonstandard Model ◽

Standard Part ◽

The Standard Model ◽

Nonstandard Models ◽

Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

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Some highly saturated models of Peano arithmetic

Journal of Symbolic Logic ◽

10.2178/jsl/1190150284 ◽

2002 ◽

Vol 67 (4) ◽

pp. 1265-1273

Author(s):

James H. Schmerl

Keyword(s):

Regular Cardinal ◽

Peano Arithmetic ◽

Saturated Model ◽

Elementary Extension ◽

Saturated Models ◽

The One ◽

Recursively Saturated

Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal and is a λ-saturated model of PA such that ∣M∣ > λ, then has an elementary extension of the same cardinality which is also λ-saturated and which, in addition, is rather classless. The construction in [10] produced a model for which cf() = λ+. We asked in Question 5.1 of [10] what other cofinalities could such a model have. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.

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METAVALUATIONS

Bulletin of Symbolic Logic ◽

10.1017/bsl.2017.29 ◽

2017 ◽

Vol 23 (3) ◽

pp. 296-323 ◽

Cited By ~ 1

Author(s):

ROSS T. BRADY

Keyword(s):

Standard Model ◽

Model Theory ◽

Liar Paradox ◽

Peano Arithmetic ◽

Theoretic Approach ◽

Relevant Logics ◽

Theoretic Proof ◽

Image Position ◽

The Liar ◽

The Liar Paradox

AbstractThis is a general account of metavaluations and their applications, which can be seen as an alternative to standard model-theoretic methodology. They work best for what are called metacomplete logics, which include the contraction-less relevant logics, with possible additions of Conjunctive Syllogism, (A→B) & (B→C) → .A→C, and the irrelevant, A→ .B→A, these including the logic MC of meaning containment which is arguably a good entailment logic. Indeed, metavaluations focus on the formula-inductive properties of theorems of entailment form A→B, splintering into two types, M1- and M2-, according to key properties of negated entailment theorems (see below). Metavaluations have an inductive presentation and thus have some of the advantages that model theory does, but they represent proof rather than truth and thus represent proof-theoretic properties, such as the priming property, if ├ A $\vee$ B then ├ A or ├ B, and the negated-entailment properties, not-├ ∼(A→B) (for M1-logics, with M1-metavaluations) and ├ ∼(A→B) iff ├ A and ├ ∼ B (for M2-logics, with M2-metavaluations). Topics to be covered are their impact on naive set theory and paradox solution, and also Peano arithmetic and Godel’s First and Second Theorems. Interesting to note here is that the familiar M1- and M2-metacomplete logics can be used to solve the set-theoretic paradoxes and, by inference, the Liar Paradox and key semantic paradoxes. For M1-logics, in particular, the final metavaluation that is used to prove the simple consistency is far simpler than its correspondent in the model-theoretic proof in that it consists of a limit point of a single transfinite sequence rather than that of a transfinite sequence of such limit points, as occurs in the model-theoretic approach. Additionally, it can be shown that Peano Arithmetic is simply consistent, using metavaluations that constitute finitary methods. Both of these results use specific metavaluational properties that have no correspondents in standard model theory and thus it would be highly unlikely that such model theory could prove these results in their final forms.

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Large discrete parts of the E-tree

Journal of Symbolic Logic ◽

10.2307/2274587 ◽

1988 ◽

Vol 53 (3) ◽

pp. 980-984 ◽

Cited By ~ 3

Author(s):

Harold Simmons

Keyword(s):

Standard Model ◽

Peano Arithmetic ◽

Unique Root ◽

First Order ◽

Aronszajn Tree ◽

The Standard Model

Let PA be first order Peano arithmetic, let Λ be the lattice of Π1 sentences modulo PA, and let S be the poset of prime filters of Λ ordered by reverse inclusion. We show there are large convex discrete parts of S; in particular there are convex parts which form a completed Baire tree or an Aronszajn tree.The elements of S, which we call nodes, correspond to the extensions of PA which are complete for sentences. Equivalently, for each model of PA the Π1-theory ∀() of is a node, and every node occurs in this form. Note that the Π1-theory ∀() of the standard model (i.e. the filter of true Π1 sentences) is the unique root of S.This poset S, which is sometimes called the E-tree, was first studied in [1] where it is shown that:(1) The poset is tree-like, i.e. the set of predecessors of any node is linearly ordered.(2) The poset has branches, each of which is closed under unions and intersections; in particular each branch has a maximum member.(3) There are branches on which ∀() does not have an immediate successor. Further properties of the E-tree are given in [2]−[7]. In particular in [4] Misercque shows that:(4) There are branches on which ∀() does have an immediate successor.(5) There are nodes with both an immediate predecessor and an immediate successor.The two results (3) and (4) show that there are fundamentally different branches of S, and (5) shows that parts of branches may be discrete.

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Conservative extensions of models of set theory and generalizations

Journal of Symbolic Logic ◽

10.2307/2273912 ◽

1986 ◽

Vol 51 (4) ◽

pp. 1005-1021 ◽

Cited By ~ 1

Author(s):

Ali Enayat

Keyword(s):

Set Theory ◽

Satisfying Condition ◽

Peano Arithmetic ◽

Negative Answer ◽

Compactness Theorem ◽

Elementary Extension ◽

Direct Contrast ◽

Conservative Extensions ◽

Models Of Set Theory ◽

Completeness Theorems

An attempt to answer the following question gave rise to the results of the present paper. Let be an arbitrary model of set theory. Does there exist an elementary extension of satisfying the two requirements: (1) contains an ordinal exceeding all the ordinals of ; (2) does not enlarge any (hyper) integer of ? Note that a trivial application of the ordinary compactness theorem produces a model satisfying condition (1); and an internal ultrapower modulo an internal ultrafilter produces a model satisfying condition (2) (but not (1), because of the axiom of replacement). Also, such a satisfying both conditions (1) and (2) exists if the external cofinality of the ordinals of is countable, since by [KM], would then have an elementary end extension.Using a class of models constructed by M. Rubin using in [RS], and already employed in [E1], we prove that our question in general has a negative answer (see Theorem 2.3). This result generalizes the results of M. Kaufmann and the author (appearing respectively in [Ka] and [E1]) concerning models of set theory with no elementary end extensions.In the course of the proof it was necessary to establish that all conservative extensions (see Definition 2.1) of models of ZF must be cofinal. This is in direct contrast with the case of Peano arithmetic where all conservative extensions are end extensional (as observed by Phillips in [Ph1]). This led the author to introduce two useful weakenings of the notion of a conservative end extension which, as shown by the “completeness” theorems in §3, can exist.

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On the complexity of models of arithmetic

Journal of Symbolic Logic ◽

10.2307/2273150 ◽

1982 ◽

Vol 47 (2) ◽

pp. 403-415 ◽

Cited By ~ 14

Author(s):

Kenneth McAloon

Keyword(s):

Standard Model ◽

Initial Segment ◽

Peano Arithmetic ◽

Nonstandard Model ◽

Recursively Enumerable ◽

The Standard Model ◽

Models Of Arithmetic

AbstractLet P0 be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let M be a countable, nonstandard model of P0 whose domain we suppose to be the standard integers. Let T be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment M′ of M which is a model of T such that the complete diagram of M′ is Turing reducible to the atomic diagram of M. Moreover, neither the addition nor the multiplication of M is recursive.

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Model-theoretic properties characterizing Peano arithmetic

Journal of Symbolic Logic ◽

10.2307/2275063 ◽

1991 ◽

Vol 56 (3) ◽

pp. 949-963 ◽

Cited By ~ 1

Author(s):

Richard Kaye

Keyword(s):

Peano Arithmetic ◽

Countable Model ◽

Elementary Extension ◽

First Order ◽

Complete Extension ◽

Order Language

AbstractLet = {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order -theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying(i) K has no proper elementary substructures, and(ii) whenever L ≻ K is a countable elementary extension there is and such that .Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

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