Degrees of Non-Determinacy and Game Logics on Cardinals under the
Axiom of Determinacy
Zhenhao Li
Abstract:
Blass showed that on each infinite cardinal, there is an algebra
structure of games on it. Blass defined a reducibility relation on
games via which he classified games into degrees of non-determinacy
and proved nice properties of the degree structures on certain
cardinals using the axiom of choice. Later Blass gave a game semantics
to affine logic, an extension of linear logic, using his game
algebra. He proved this game semantics is consistent (sound) but not
complete. But he proved two nice completeness theorem for fragments of
affine logic using the axiom of choice.
This thesis gives a detailed exposition of Blassâ€™s work on degree of
non-determinacy and game semantics of linear logic, with an emphasis
on the roles of cardinals and the usage of the axiom of choice, and
contains our studies of degrees of non-determinacy and game logics on
infinite cardinals in a set theory system without using the axiom of
choice, namely in ZF with the axiom of determinacy.