Factorization of the Shoenfield-like Bounded Functional Interpretation

Abstract

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].

A functional interpretation for nonstandard arithmetic

“…We will work out the technical details in Sections 6 and 7 below. The resulting functional interpretations have some striking similarities with the bounded functional interpretations introduced by Ferreira and Oliva in [12] and [10] (see also [13]). In the same way, Herbrand realizability seems related to the bounded modified realizability interpretation due to Ferreira and Nunes (for which, see [11]).…”

A parametrised functional interpretation of Heyting arithmetic

“…For example, as shown in [1,29], Jean-Louis Krivine's negative translation is the correct tool to connect Gödel's Dialectica with Shoenfield's interpretation. Other factorisations were obtained in [5,14,29,27]. It is our impression that composing our intuitionistic parametrised interpretation with various negative translations would entail parametrised classical interpretations that allows one to obtain all the standard interpretations for classical logic, showing factorisations are a general feature among functional interpretations.…”

Parametrised Functional Interpretations

“…For example, as shown in [1,28], Jean-Louis Krivine's negative translation is the correct tool to connect Gödel's Dialectica with Shoenfield's interpretation. Other factorisations were obtained in [5,13,28,26]. It is our impression that composing our intuitionistic parametrised interpretation with various negative translations would entail parametrised classical interpretations that allows one to obtain all the standard interpretations for classical logic, showing factorisations are a general feature among functional interpretations.…”