The countable versus uncountable branching recurrences in computability logic

TL;DR

This paper compares countable and uncountable branching recurrences in Computability Logic, introducing a simplified version, proving their equivalence, and analyzing their logical relationships and strengths.

Contribution

It introduces a simplified countable branching recurrence, proves its equivalence to the original, and clarifies the logical hierarchy between countable and uncountable recurrences.

Findings

The simplified countable recurrence is equivalent to the original.

The logic induced by the countable recurrence is a proper superset of that by the uncountable recurrence.

Uncountable recurrence is strictly stronger than countable recurrence.

Abstract

This paper introduces a new simplified version of the countable branching recurrence of Computability Logic, proves its equivalence to the old one, and shows that the basic logic induced by it is a proper superset of the basic logic induced by the uncountable branching recurrence. A further result of this paper is showing that the countable branching recurrence is strictly weaker than the uncountable branching recurrence in the sense that the latter logically implies the former but not vice versa.