Arithmetic translations of axiom systems

Complete independence of an axiom system for central translations

A recently proposed axiom system for André’s central translation structures is improved upon. First, one of its axioms turns out to be dependent (derivable from the other axioms). Without this axiom, the axiom system is indeed independent. Second, whereas most of the original independence models were infinite, finite independence models are available. Moreover, the independence proof for one of...

A Computational Interpretation of the Axiom of Determinacy in Arithmetic

We investigate the computational content of the axiom of determinacy (AD) in the setting of classical arithmetic in all finite types with the principle of dependent choices (DC). By employing the notion of realizability interpretation for arithmetic given by Berardi, Bezem and Coquand (1998), we interpret the negative translation of AD. Consequently, the combination of the negative translation ...

Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice

Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, e...

Mechanising Set Theory: Cardinal Arithmetic and the Axiom of Choice

On Axiom Systems of Pseudo-BCK Algebras

A symplified axiomatization of pseudo-BCK algebras is given. In addition, we obtain a system of axioms defining bounded commutative pseudoBCK algebras and show that these axioms are independent (that is, none follows from the others). 2010 Mathematics Subject Classification: 03G25, 06F05, 06F35