The countable versus uncountable branching recurrences in computability logic

This paper introduces a new simplified version of the countable branching recurrence ◦ |א0 of Computability Logic, proves its equivalence to the old one, and shows that the basic logic induced by ◦ |א0 (i.e., the one in the signature {¬,∧,∨, ◦ |א0 , ◦| א}) is a proper superset of the basic logic induced by the uncountable branching recurrence ◦ | (i.e., the one in the signature {¬,∧,∨, ◦ | , ◦| }). A further result of this paper is showing that ◦ |א0 is strictly weaker than ◦ | in the sense that ◦ | F logically implies ◦ |א0 F but not vice versa. MSC: primary: 03B47; secondary: 03B70; 68Q10; 68T27; 68T15.