Subsystems of Second Order Arithmetic

2009 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Second Order ◽  
Second Order Arithmetic ◽  
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

Reverse Mathematics: The Playground of Logic

2010 ◽  
Vol 16 (3) ◽  
pp. 378-402 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Reverse Mathematics ◽  
Computational Approach ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Base Theory ◽  
Order Arithmetic

AbstractThis paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

The Baire category theorem in weak subsystems of second-order arithmetic

1993 ◽  
Vol 58 (2) ◽  
pp. 557-578 ◽  
Author(s):  
Douglas K. Brown ◽  
Stephen G. Simpson
Keyword(s):  
Functional Analysis ◽  
Model Theory ◽  
Baire Category ◽  
Second Order ◽  
Base System ◽  
Baire Category Theorem ◽  
Second Order Arithmetic ◽  
Category Theorem ◽  
Order Arithmetic

AbstractWorking within weak subsystems of second-order arithmetic Z2 we consider two versions of the Baire Category theorem which are not equivalent over the base system RCA0. We show that one version (B.C.T.I) is provable in RCA0 while the second version (B.C.T.II) requires a stronger system. We introduce two new subsystems of Z2, which we call and , and , show that suffices to prove B.C.T.II. Some model theory of and its importance in view of Hilbert's program is discussed, as well as applications of our results to functional analysis.

scholarly journals Nonstandard second-order arithmetic and Riemannʼs mapping theorem

2014 ◽  
Vol 165 (2) ◽  
pp. 520-551 ◽  
Author(s):  
Yoshihiro Horihata ◽  
Keita Yokoyama
Keyword(s):  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic
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