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

Journal of Symbolic Logic 59 (1):113-123 (1994)
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Abstract

Let 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 Steffens [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 ART0. 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

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Citations of this work

Formalizing forcing arguments in subsystems of second-order arithmetic.Jeremy Avigad - 1996 - Annals of Pure and Applied Logic 82 (2):165-191.
On the relationship between ATR 0 and.Jeremy Avigad - 1996 - Journal of Symbolic Logic 61 (3):768-779.
Reverse mathematics: the playground of logic.Richard A. Shore - 2010 - Bulletin of Symbolic Logic 16 (3):378-402.
A model-theoretic approach to ordinal analysis.Jeremy Avigad & Richard Sommer - 1997 - Bulletin of Symbolic Logic 3 (1):17-52.

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References found in this work

Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.
Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.
Borel quasi-orderings in subsystems of second-order arithmetic.Alberto Marcone - 1991 - Annals of Pure and Applied Logic 54 (3):265-291.

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