Room 327

Non-classical set theories and logics associated with them

Sourav Tarafder

Calcutta University

Non-classical logics and non-classical set theories
constitute a growing research area. Intuitionistic set theory, quantum set theory,
paraconsistent set theory, many-valued set theory are examples of non-classical set theories. By a non-classical set theory we shall mean a set theory whose underlying logic is non-classical. We are specially interested in finding a paraconsistent set theory. We have generalised the Boolean-valued model
construction of classical set theory to a generalised algebra-valued model. An algebra called 'reasonable implication algebra' will be introduced which can be used for producing an algebra-valued model of a non-classical set theory. A three-valued algebra PS3 will be shown as an interpretation of reasonable implication algebra which is neither a Boolean algebra nor a Heyting algebra. It is proved that the logic LPS3 which is sound and (weak) complete with respect to PS3 is paraconsistent. Then we shall focus on the paraconsistent set theory whose underlying logic is the predicate extension of LPS3. The algebra-valued model VPS3 of this paraconsistent set theory will be discussed, especially the ordinals in VPS3. It will be shown that VPS3 does not satisfy the
Leibniz's law of indiscernibility of identicals.