Calliope Ryan-Smith (University of Leeds)

Finality of forcing

Iterated forcing is a powerful tool for ouroboric arguments in set theory that rely on repeatedly creating or destroying some property until your construction eats its own tail and gives you your final result (in fact a similar argument may be applied to many ideas in set theory, especially when ordinals are involved. A simple example would be the \omega_1th stage of the Borel/projective hierarchies being no more than the union of their prior stages). To this end, it is often a helpful feature of an iterated notion of forcing that in the final model one has not introduced any new reals (subsets of \kappa, functions \lambda\to\lambda, etc) that are not already present in some intermediate stage. This behaviour is precisely captured by finality, which we shall define and give an exact characterisation of.