33 research outputs found
A functional interpretation for nonstandard arithmetic
We introduce constructive and classical systems for nonstandard arithmetic
and show how variants of the functional interpretations due to Goedel and
Shoenfield can be used to rewrite proofs performed in these systems into
standard ones. These functional interpretations show in particular that our
nonstandard systems are conservative extensions of extensional Heyting and
Peano arithmetic in all finite types, strengthening earlier results by
Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting
algorithm can be used for term extraction purposes. To conclude the paper, we
will point out some open problems and directions for future research and
mention some initial results on saturation principles
The strength of countable saturation
We determine the proof-theoretic strength of the principle of countable
saturation in the context of the systems for nonstandard arithmetic introduced
in our earlier work.Comment: Corrected typos in Lemma 3.4 and the final paragraph of the
conclusio
The Herbrand Topos
We define a new topos, the Herbrand topos, inspired by the modified
realizability topos and our earlier work on Herbrand realizability. We also
introduce the category of Herbrand assemblies and characterise these as the
double-negation-separated objects in the Herbrand topos. In addition, we show
that the category of sets is included as the category of
double-negation-sheaves and prove that the inclusion functor preserves and
reflects validity of first-order formulas