Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of (...) the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models. (shrink)

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Gödel 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 E-HAω and E-PAω, strengthening earlier results by Moerdijk and Palmgren, and 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, including some initial results on saturation principles. (shrink)

In view of an enhancement of our implementation on the computer, we explore the possibility of an algorithmic optimization of the various proof-theoretic techniques employed by Kohlenbach for the synthesis of new effective uniform bounds out of established qualitative proofs in Numerical Functional Analysis. Concretely, we prove that the method of “colouring” some of the quantifiers as “non-computational” extends well to ε-arithmetization, elimination-of-extensionality and model-interpretation.

We upgrade the light Dialectica interpretation [6] by adding two more light universal quantifiers, which are both semi-computational and semi-uniform and complement each other. An illustrative example is presented for the new light quantifiers and a new application is given for the older uniform quantifier. The realizability of new light negative formulations for the Axiom of Choice and for the Independence of Premises is explored in the new setting.