A model of peano arithmetic with no elementary end extension

1978 ◽  
Vol 43 (3) ◽  
pp. 563-567 ◽  
Author(s):  
George Mills
Keyword(s):  
Peano Arithmetic ◽  
Nonstandard Model

AbstractWe 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.

Two theorems on degrees of models of true arithmetic

1984 ◽  
Vol 49 (2) ◽  
pp. 425-436 ◽  
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.

Infinite substructure lattices of models of Peano Arithmetic

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.

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
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.

Uniqueness, collection, and external collapse of cardinals in IST and models of Peano arithmetic

1995 ◽  
Vol 60 (1) ◽  
pp. 318-324 ◽  
Author(s):  
V. Kanovei
Keyword(s):  
Set Theory ◽  
Peano Arithmetic ◽  
General Technique ◽  
Nonstandard Model ◽  
Internal Set

AbstractWe 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.

ILLUSORY MODELS OF PEANO ARITHMETIC

2016 ◽  
Vol 81 (3) ◽  
pp. 1163-1175 ◽  
Author(s):  
MAKOTO KIKUCHI ◽  
TAISHI KURAHASHI
Keyword(s):  
Initial Segment ◽  
Peano Arithmetic ◽  
Nonstandard Model

AbstractBy using a provability predicate of PA, we define ThmPA(M) as the set of theorems of PA in a model M of PA. We say a model M of PA is (1) illusory if ThmPA(M) ⊈ ThmPA(ℕ), (2) heterodox if ThmPA(M) ⊈ TA, (3) sane if M ⊨ ConPA, and insane if it is not sane, (4) maximally sane if it is sane and ThmPA(M) ⊆ ThmPA(N) implies ThmPA(M) = ThmPA(N) for every sane model N of PA. We firstly show that M is heterodox if and only if it is illusory, and that ThmPA(M) ∩ TA ≠ ThmPA(ℕ) for any illusory model M. Then we show that there exists a maximally sane model, every maximally sane model satisfies ¬ConPA+ConPA, and there exists a sane model of ¬ConPA+ConPA which is not maximally sane. We define that an insane model is (5) illusory by nature if its every initial segment being a nonstandard model of PA is illusory, and (6) going insane suddenly if its every initial segment being a sane model of PA is not illusory. We show that there exists a model of PA which is illusory by nature, and we prove the existence of a model of PA which is going insane suddenly.

On certain types and models for arithmetic

1974 ◽  
Vol 39 (1) ◽  
pp. 151-162 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Model Theory ◽  
Peano Arithmetic ◽  
Minimal Extension ◽  
Nonstandard Model ◽  
Conservative Extensions

AbstractThere 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 model-theoretic 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.

Provability of the pigeonhole principle and the existence of infinitely many primes

1988 ◽  
Vol 53 (4) ◽  
pp. 1235-1244 ◽  
Author(s):  
J. B. Paris ◽  
A. J. Wilkie ◽  
A. R. Woods
Keyword(s):  
Positive Solution ◽  
Peano Arithmetic ◽  
Pigeonhole Principle ◽  
Integer Part ◽  
Know How ◽  
Nonstandard Model ◽  
The Difference ◽  
Image Position

In this note we shall be interested in the following problems.Problem 1. Can IΔ0 ⊢ ∀x∃y > x(y is prime)?Here I Δ0 is Peano arithmetic with the induction axiom restricted to bounded (i.e. Δ0) formulae.Problem 2. Can IΔ0 ⊢ Δ0 PHP?Here Δ0 PHP (Δ0 pigeonhole principle) is the schemafor θ ∈ Δ0, or equivalently in IΔ0, for a Δ0 formula F(x,y)written .By obtaining partial solutions to Problem 2 we shall show that Problem 1 has a positive solution if IΔ0 is replaced by IΔ0 + ∀xxlog(x) exists.Our notation will be entirely standard (see for example [3] and [4]). In particular all logarithms will be to the base 2 and in expressions like log(x), (1 + ε)x, etc. we shall always mean the integer part of these quantities.Concerning Problem 2 we remark that it is shown in [5] that for k ∈ N and F ∈ Δ0,As far as we know this is the best result of this form, in that we do not know how to replace log(z)k by anything larger. However, as we shall show in Theorem 1, we can do much better if we increase the difference between the sizes of the domain and range of F.In what follows let M be a countable nonstandard model of IΔ0, and let be those subsets of M defined by Δ0 formulae with parameters from M.Theorem 1. For k ∈ N andF ∈ Δ0,Here log0(x) = x, logk + 1(x) = log(logk(x)).Proof. To simplify matters, consider first the case k = 1. So assume M ⊨ alog(a) exists and with and a > 1. The idea of the proof is the following.

Torre models in the isols

1994 ◽  
Vol 59 (1) ◽  
pp. 140-150 ◽  
Author(s):  
Joseph Barback
Keyword(s):  
Number Theory ◽  
Structural Property ◽  
Recursive Function ◽  
Peano Arithmetic ◽  
The Other ◽  
Close Connection ◽  
New Variety ◽  
Nonstandard Model ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractIn [14] J. Hirschfeld established the close connection of models of the true AE sentences of Peano Arithmetic and homomorphic images of the semiring of recursive functions. This fragment of Arithmetic includes most of the familiar results of classical number theory. There are two nice ways that such models appear in the isols. One way was introduced by A. Nerode in [20] and is referred to in the literature as Nerode Semirings. The other way is called a tame model. It is very similar to a Nerode Semiring and was introduced in [6]. The model theoretic properties of Nerode Semirings and tame models have been widely studied by T. G. McLaughlin ([16], [17], and [18]).In this paper we introduce a new variety of tame model called a torre model. It has as a generator an infinite regressive isol with a nice structural property relative to recursively enumerable sets and their extensions to the isols. What is then obtained is a nonstandard model in the isols of the fragment of Peano Arithmetic with the following property: Let T be a torre model. Let f be any recursive function, and let fΛ be its extension to the isols. If there is an isol A with fΛ(A) ϵ T, then there is also an isol B ϵ T with fΛ(B) = fΛ(A).

Full Satisfaction Classes and Recursive Saturation

1981 ◽  
Vol 24 (3) ◽  
pp. 295-297 ◽  
Author(s):  
A. H. Lachlan
Keyword(s):  
Peano Arithmetic ◽  
Recursive Saturation ◽  
Nonstandard Model ◽  
Recursively Saturated ◽  
Satisfaction Classes ◽  
Satisfaction Class

AbstractIt is shown that a nonstandard model of Peano arithmetic which has a full satisfaction class is necessarily recursively saturated.

Real closed fields and models of Peano arithmetic

2012 ◽  
Vol 75 (1) ◽  
pp. 1-11 ◽  
Author(s):  
P. D'Aquino ◽  
J. F. Knight ◽  
S. Starchenko
Keyword(s):  
Peano Arithmetic ◽  
Integer Part ◽  
The Real ◽  
Real Closed Fields ◽  
Nonstandard Model ◽  
Ordered Field ◽  
Closed Fields ◽  
Real Closure ◽  
Ordered Ring ◽  
Recursively Saturated

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.

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