"classical" computability theory and the category of Sets

Hello Cat Theory list, Please be gentle. In the past I studied computability theory. It seems to me that this theory is built on the category of Sets(elementary topos), i.e. this computability theory assumes using classical logic with LEM and boolean subobject classifier for concepts like semi-decidability, etc. Is there a notion of intuitionistic computability theory built on other topoi where LEM is absent from the accompanying higher logic and the topos' subobject classifier has a internal Heyting algebra(that is not boolean)?? Is this what realizibility delves into(I have yet to study realizibility concepts). Kind regards, Vasya [For admin and other information see: http://www.mta.ca/~cat-dist/ ]