A herbrandized functional interpretation of classical first-order logic

We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.info:eu-repo/semantics/publishedVersio

Reverse Mathematics and parameter-free Transfer

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi

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

Functional interpretations and applications

Functional interpretations are maps of formulas from the language of one theory into the language of another theory, in such a way that provability is preserved. These interpretations typically replace logical relations by functional relations. Functional interpretations have many uses, such as relative consistency results, conservation results, and extraction of computational content from proofs as is the case in the so-called proof mining program. I will present several recent functional interpretations and some results that come from these interpretations. I will also give examples of application of functional interpretations, in the spirit of the proof mining program

Grilliot's trick in Nonstandard Analysis

The technique known as Grilliot's trick constitutes a template for explicitly defining the Turing jump functional $(\exists^2)$ in terms of a given effectively discontinuous type two functional. In this paper, we discuss the standard extensionality trick: a technique similar to Grilliot's trick in Nonstandard Analysis. This nonstandard trick proceeds by deriving from the existence of certain nonstandard discontinuous functionals, the Transfer principle from Nonstandard analysis limited to $\Pi_1^0$-formulas; from this (generally ineffective) implication, we obtain an effective implication expressing the Turing jump functional in terms of a discontinuous functional (and no longer involving Nonstandard Analysis). The advantage of our nonstandard approach is that one obtains effective content without paying attention to effective content. We also discuss a new class of functionals which all seem to fall outside the established categories. These functionals directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page