A synthetic axiomatization of Map Theory

Chantal Berline (1) , Klaus Grue (2)

1 PPS
2 DIKU - Department of Computer Science [Copenhagen]

This paper presents a subtantially simplified axiomatization of Map Theory and proves the consistency of this axiomatization in ZFC under the assumption that there exists an inaccessible ordinal. Map Theory axiomatizes lambda calculus plus Hilbert's epsilon operator. All theorems of ZFC set theory including the axiom of foundation are provable in Map Theory, and if one omits Hilbert's epsilon operator from Map Theory then one is left with a computer programming language. Map Theory fulfills Church's original aim of introducing lambda calculus. Map Theory is suited for reasoning about classical mathematics as well ascomputer programs. Furthermore, Map Theory is suited for eliminating thebarrier between classical mathematics and computer science rather than just supporting the two fields side by side. Map Theory axiomatizes a universe of "maps", some of which are "wellfounded". The class of wellfounded maps in Map Theory corresponds to the universe of sets in ZFC. The first version MT0 of Map Theory had axioms which populated the class of wellfounded maps, much like the power set axiom et.al. populates the universe of ZFC. The new axiomatization MT of Map Theory is "synthetic" in the sense that the class of wellfounded maps is defined inside MapTheory rather than being introduced through axioms. In the paper we define the notion of kappa- and kappasigma-expansions and prove that if sigma is the smallest strongly inaccessible cardinal then canonical kappasigma expansions are models of MT (which proves the consistency). Furthermore, in the appendix, we prove that canonical omega-expansions are fully abstract models of the computational part of Map Theory.

Mots clés: Lambda-calculus; Foundations of mathematics; Map Theory; Hilbert's epsilon operator; Kappa-Scott semantics

Domaines: Logique [math.LO]; Logique en informatique [cs.LO]

Chantal Berline, Klaus Grue. A synthetic axiomatization of Map Theory. Theoretical Computer Science, 2016, 614, pp.1-62. ⟨10.1016/j.tcs.2015.11.028⟩. ⟨hal-00678410v3⟩