On the strength of König's duality theorem for countable bipartite graphs

1994 ◽  
Vol 59 (1) ◽  
pp. 113-123 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Bipartite Graph ◽  
Duality Theorem ◽  
Bipartite Graphs ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Vertex Covering ◽  
Order Arithmetic ◽  
Countably Infinite ◽  
Transfinite Recursion

AbstractLet CKDT be the assertion that for every countably infinite bipartite graph G, there exist a vertex covering C of G and a matching M in G such that C consists of exactly one vertex from each edge in M. (This is a theorem of Podewski and Stefifens [12].) Let ATR0 be the subsystem of second-order arithmetic with arithmetical transfinite recursion and restricted induction. Let RCA0 be the subsystem of second-order arithmetic with recursive comprehension and restricted induction. We show that CKDT is provable in ATR0. Combining this with a result of Aharoni, Magidor, and Shore [2], we see that CKDT is logically equivalent to the axioms of ATR0, the equivalence being provable in RCA0.

scholarly journals THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

2017 ◽  
Vol 10 (2) ◽  
pp. 357-396 ◽  
Author(s):  
WALTER DEAN ◽  
SEAN WALSH
Keyword(s):  
Large Scale ◽  
Second Order ◽  
Descriptive Set Theory ◽  
Second Order Arithmetic ◽  
Mathematics Program ◽  
Order Arithmetic ◽  
Transfinite Recursion ◽  
Descriptive Set ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractThis paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König’s Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

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.

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