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 FIPP₁. That turned out to not be the proper formulation and so we proposed an alternative version FIPP₂. Tao himself formulated yet another version FIPP₃ in a revised version of his essay. We give (...) a counterexample to FIPP₁ and discuss for both of the versions FIPP₂ and FIPP₃ the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP₂ and IPP ↔ FIPP₃ in the context of reverse mathematics ([9]). 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. (shrink)

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.

We refute the conjecture that all negative translations are intuitionistically equivalent by giving two counterexamples. Then we characterise the negative translations intuitionistically equivalent to the usual ones.

La logique classique (la logique des mathématiques non-constructives) est plus forte que la logique intuitionniste (la logique des mathématiques constructives). Malgré cela, il existe des copies de la logique classique dans la logique intuitionniste. Toutes les copies habituellement trouvées dans la littérature sont les mêmes. Ce qui soulève la question suivante : la copie est-elle unique? Nous répondons négativement en présentant trois copies différentes.

This 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 tobe 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. (shrink)