The strength of nonstandard methods in arithmetic

1984 ◽  
Vol 49 (4) ◽  
pp. 1039-1058 ◽  
Author(s):  
C. Ward Henson ◽  
Matt Kaufmann ◽  
H. Jerome Keisler
Keyword(s):  
Initial Segment ◽  
Nonstandard Analysis ◽  
Peano Arithmetic ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Natural Analogues ◽  
Order Arithmetic

AbstractWe 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.

scholarly journals Multi-sorted version of second order arithmetic

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Farida Kachapova
Keyword(s):  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Second Order ◽  
Natural Numbers ◽  
Natural Form ◽  
Second Order Arithmetic ◽  
Arithmetical Truth ◽  
Order Arithmetic

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

An infinitistic rule of proof

1968 ◽  
Vol 32 (4) ◽  
pp. 447-451 ◽  
Author(s):  
H. B. Enderton
Keyword(s):  
Formal System ◽  
Peano Arithmetic ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic

In this paper we consider a fonnal system of second-order Peano arithmetic with a rule of inference stronger than the ω-rule [3]. We also consider the relation to a class of models for analysis (i.e. second-order arithmetic) which lies between the class of ω-models and the class of β-models [5].The notation used is largely that of [3] and [5]. We assume that the reader has some familiarity with at least the ideas of the former. The formal system (A) of Peano arithmetic employed in [3] includes the comprehension axioms and the second-order induction axiom.

On the strength of nonstandard analysis

1986 ◽  
Vol 51 (2) ◽  
pp. 377-386 ◽  
Author(s):  
C. Ward Henson ◽  
H. Jerome Keisler
Keyword(s):  
Set Theory ◽  
Nonstandard Analysis ◽  
Mathematical Practice ◽  
Second Order ◽  
Erroneous Conclusion ◽  
Second Order Arithmetic ◽  
Correct Statement ◽  
Classical Mathematics ◽  
Order Arithmetic ◽  
Almost All

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis.The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes.

MINIMAL TRUTH AND INTERPRETABILITY

2009 ◽  
Vol 2 (4) ◽  
pp. 799-815 ◽  
Author(s):  
MARTIN FISCHER
Keyword(s):  
Peano Arithmetic ◽  
Second Order ◽  
Conservative Extension ◽  
Axiomatic Theories Of Truth ◽  
Second Order Arithmetic ◽  
Theories Of Truth ◽  
Fine Grained ◽  
Order Arithmetic ◽  
Weak Theories

In this paper we will investigate different axiomatic theories of truth that are minimal in some sense. One criterion for minimality will be conservativity over Peano Arithmetic. We will then give a more fine-grained characterization by investigating some interpretability relations. We will show that disquotational theories of truth, as well as compositional theories of truth with restricted induction are relatively interpretable in Peano Arithmetic. Furthermore, we will give an example of a theory of truth that is a conservative extension of Peano Arithmetic but not interpretable in it. We will then use stricter versions of interpretations to compare weak theories of truth to subsystems of second-order arithmetic.

scholarly journals Short note: Least fixed points versus least closed points

2021 ◽  
Author(s):  
Gerhard Jäger
Keyword(s):  
Fixed Points ◽  
Short Note ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Arithmetic Operator

AbstractThis short note is on the question whether the intersection of all fixed points of a positive arithmetic operator and the intersection of all its closed points can proved to be equivalent in a weak fragment of second order arithmetic.

UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

2014 ◽  
Vol 79 (4) ◽  
pp. 1001-1019 ◽  
Author(s):  
ASHER M. KACH ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Direct Product ◽  
Disjoint Union ◽  
Second Order ◽  
Linear Orders ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Natural Classes ◽  
Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

scholarly journals Periodic points and subsystems of second-order arithmetic

1993 ◽  
Vol 62 (1) ◽  
pp. 51-64 ◽  
Author(s):  
Harvey Friedman ◽  
Stephen G. Simpson ◽  
Xiaokang Yu
Keyword(s):  
Periodic Points ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic
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