On Diophantine equations solvable in models of open induction

1990 ◽  
Vol 55 (2) ◽  
pp. 779-786 ◽  
Author(s):  
Margarita Otero
Keyword(s):  
Diophantine Equations ◽  
Main Lemma ◽  
Peano Arithmetic ◽  
The Real ◽  
Real Closure ◽  
Open Induction ◽  
Induction Scheme

AbstractWe consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas.We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣f(x,y) ∣ < 1 for every (x,y) Є C, where C is the curve defined on the real closure of M by C: x2 + y2 = a and a > 0 is a nonstandard element of M.

Open induction and the true theory of rationals

1987 ◽  
Vol 52 (3) ◽  
pp. 793-801
Author(s):  
Zofia Adamowicz
Keyword(s):  
Linear Space ◽  
Usual Sense ◽  
Saturated Model ◽  
The Real ◽  
Maximal Subset ◽  
True Theory ◽  
Real Closure ◽  
Open Induction ◽  
Infinite Sets ◽  
Image Position

In the paper we prove the following theorem:Theorem. There is a model N of open induction in which the set of primes is bounded and N is such that its field of fractions 〈N*, +, ·, <〉 is elementarily equivalent to 〈Q, +, ·, <〉 (the standard rationals).We fix an ω1-saturated model 〈M, +, ·, <〉 of PA. Let 〈M*, +, ·, <〉 denote the field of fractions of M. The model N that we are looking for will be a substructure of 〈M*, +, ·, <〉.If A ⊆ M* then let Ā denote the ring generated by A within M*, Ậ the real closure of A, and A* the field of fractions generated by A. We haveLet J ⊆ M. Then 〈M*, +, ·〉 is a linear space over J*. If x1,…,xk ∈ M*, we shall say that x1,…,xk are J-independent if 〈1, x1,…, xk〉 are J*-independent in the usual sense. As usual, we extend the notion of J-independence to the case of infinite sets.If A ⊆ M* and X ⊆ A, then we say that X is a J-basis of A if X is a maximal subset of A which is J-independent.Definition 1.1. By a J-form ρ we mean a function from (M*)k into M*, of the formwhere q0,…, qk ∈ J*If υ ∈ M, we say that ρ is a υ-form if the numerators and denominators of the qi's have absolute values ≤ υ.

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.

scholarly journals NOTES ON BOUNDED INDUCTION FOR THE COMPOSITIONAL TRUTH PREDICATE

2017 ◽  
Vol 10 (3) ◽  
pp. 455-480 ◽  
Author(s):  
BARTOSZ WCISŁO ◽  
MATEUSZ ŁEŁYK
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Truth Predicate ◽  
Base Theory ◽  
Modified Theory ◽  
Induction Scheme

AbstractWe prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ0-induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

Peano arithmetic may not be interpretable in the monadic theory of linear orders

1997 ◽  
Vol 62 (3) ◽  
pp. 848-872 ◽  
Author(s):  
Shmuel Lifsches ◽  
Saharon Shelah
Keyword(s):  
Real Line ◽  
Order Theory ◽  
Peano Arithmetic ◽  
Second Order ◽  
Linear Orders ◽  
The Real ◽  
Monadic Theory

AbstractGurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.

scholarly journals Теорема Артина для f-колец

2015 ◽  
Author(s):  
A.G. Kusraev
Keyword(s):  
Rational Functions ◽  
Sum Of Squares ◽  
Famous Theorem ◽  
The Real ◽  
Complete Ring ◽  
Positive Polynomial ◽  
Ring Of Quotients ◽  
Real Closure ◽  
F Ring ◽  
17Th Problem

The main result states that each positive polynomial p in N variables with coefficients in a unital Archimedean f-ring K is representable as a sum of squares of rational functions over the complete ring of quotients of K provided that p is positive on the real closure of K. This is proved by means of Boolean valued interpretation of Artin's famous theorem which answers Hilbert's 17th problem affirmatively.

On maximal theories

1991 ◽  
Vol 56 (3) ◽  
pp. 885-890 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Open Problem ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Finite Collection ◽  
Extension Problem ◽  
Recursive Theory ◽  
Image Position ◽  
Induction Scheme

Let S be a recursive theory. Let a theory T* consisting of Σ1 sentences be called maximal (with respect to S) if T* is maximal consistent with S, i.e. there is no Σ1 sentence consistent with T* + S which is not in T*.A maximal theory with respect to IΔ0 was considered by Wilkie and Paris in [WP] in connection with the end-extension problem.Let us recall that IΔ0 is the fragment of Peano arithmetic consisting of the finite collection of algebraic axioms PA− together with the induction scheme restricted to bounded formulas.The main open problem concerning the end-extendability of models of IΔ0 is the following:(*) Does every model of IΔ0 + BΣ1 have a proper end-extension to a model of IΔ0?Here BΣ1 is the following collection scheme:where φ runs over bounded formulas and may contain parameters.It is well known(see [KP]) that if I is a proper initial segment of a model M of IΔ0, then I satisfies IΔ0 + BΣ1.For a wide discussion of the problem (*) see [WP]. Wilkie and Paris construct in [WP] a model M of IΔ0 + BΣ1 which has no proper end-extension to a model of IΔ0 under the assumption IΔ0 ⊢¬Δ0H (see [WP] for an explanation of this assumption). Their model M is a model of a maximal theory T* where S = IΔ0.Moreover, T*, which is the set Σ1(M) of all Σ1 sentences true in M, is not codable in M.

scholarly journals Models of true arithmetic are integer parts of models of real exponentation

2021 ◽  
Vol 13 ◽  
Author(s):  
Merlin Carl ◽  
Lothar Sebastian Krapp
Keyword(s):  
Multiplicative Group ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
Integer Part ◽  
Real Numbers ◽  
The Real ◽  
Positive Elements ◽  
Closed Fields ◽  
Schanuel's Conjecture

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each model of Peano arithmetic is an integer part of a real closed field that admits an isomorphism between its ordered additive and its ordered multiplicative group of positive elements. Under the assumption of Schanuel’s Conjecture, we obtain further strengthenings for the last statement.

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