131 research outputs found
Regular Functors and Relative Realizability Categories
Relative realizability toposes satisfy a universal property that involves
regular functors to other categories. We use this universal property to define
what relative realizability categories are, when based on other categories than
of the topos of sets. This paper explains the property and gives a construction
for relative realizability categories that works for arbitrary base Heyting
categories. The universal property shows us some new geometric morphisms to
relative realizability toposes too
More on Geometric Morphisms between Realizability Toposes
Geometric morphisms between realizability toposes are studied in terms of
morphisms between partial combinatory algebras (pcas). The morphisms inducing
geometric morphisms (the {\em computationally dense\/} ones) are seen to be the
ones whose `lifts' to a kind of completion have right adjoints. We characterize
topos inclusions corresponding to a general form of relative computability. We
characterize pcas whose realizability topos admits a geometric morphism to the
effective topos.Comment: 20 page
Realizability algebras II : new models of ZF + DC
Using the proof-program (Curry-Howard) correspondence, we give a new method
to obtain models of ZF and relative consistency results in set theory. We show
the relative consistency of ZF + DC + there exists a sequence of subsets of R
the cardinals of which are strictly decreasing + other similar properties of R.
These results seem not to have been previously obtained by forcing.Comment: 28
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