We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles.The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability.We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments.1. We add the standard predicates st σ (t σ ) (for each finite type σ).2. We add the standardness axioms:(for each finite types σ and τ ).3. We add the external induction rule Φ(0) ∀x 0 (st 0 (x)→(Φ(x)→Φ(x+1))) ∀x 0 (st 0 (x)→Φ(x))
In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the "finitary" infinite pigeonhole principle FIPP 1 . That turned out to not be the proper formulation and so we proposed an alternative version FIPP 2 . Tao himself formulated yet another version FIPP 3 in a revised version of his essay.We give a counterexample to FIPP 1 and discuss for both of the versions FIPP 2 and FIPP 3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP 2 and IPP ↔ FIPP 3 in the context of reverse mathematics. In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the "big five" subsystems of second order arithmetic.
Key words Modified realizability with truth and q-variant, intuitionistic linear logic, Aczel and Kleene slashes. MSC (2010) 03F07, 03F10, 03F25This article systematically investigates so-called "truth variants" of several functional interpretations. We start by showing a close relation between two variants of modified realizability, namely modified realizability with truth and q-modified realizability. Both variants are shown to be derived from a single "functional interpretation with truth" of intuitionistic linear logic. This analysis suggests that several functional interpretations have truth and q-variants. These variants, however, require a more involved modification than the ones previously considered. Following this lead we present truth and q-variants of the Diller-Nahm interpretation, the bounded modified realizability and the bounded functional interpretation.
We adapt Streicher and Kohlenbach's proof of the factorization S = KD of the Shoenfield translation S in terms of Krivine's negative translation K and the Gödel functional interpretation D, obtaining a proof of the factorization U = KB of Ferreira's Shoenfield-like bounded functional interpretation U in terms of K and Ferreira and Oliva's bounded functional interpretation B. 1 Introduction In 1958, Gödel [5] presented a functional interpretation D of Heyting arithmetic HA ω into itself (actually, into a quantifier-free theory, for foundational reasons). When composed with a negative translation N of Peano arithmetic PA ω into HA ω (Gödel [4]), it results in a two-step functional interpretation ND of PA ω into HA ω [5]. Nine years later, Shoenfield [9] presented a one-step functional interpretation S of PA ω into HA ω . In 2007, Streicher and Kohlenbach [10], and independently Avigad [1], proved the factorization S = KD of S in terms of D and a negative translation K due to Streicher and Reus [11], inspired by Krivine [8].
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