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Reverse mathematics and the equivalence of definitions for well and better quasi-orders

Published online by Cambridge University Press:  12 March 2014

Peter Cholak ,
Alberto Marcone  and
Reed Solomon
Peter Cholak
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, Indiana 46556-4618, USA, E-mail: Peter.Cholak.1@nd.edu
Alberto Marcone
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: marcone@dimi.uniud.it
Reed Solomon
Affiliation:
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, Connecticut 06269-3009, USA, E-mail: solomon@math.uconn.edu
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Extract

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.

As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.

We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 3 , September 2004 , pp. 683 - 712
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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