Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols

Abstract

Well-known interpretations of Heyting's arithmetic of all finite types are the Diller-Nahm λ-interpretation [1] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids λ q resp. mq.

In Section 4 a chain of interpretations—called M-interpretations—is defined (it was introduced in [6], filling the “gap” between λ-interpretation and mr-interpretation.

In this paper it is shwon that it is possible to prove in one stroke the soundness resp. characterization theorems for all interpretations of HA_{ω 〈〉} (Heyting's arithmetic of all finite types with functionals for coding finite sequences). This is done by means of interpretations of systems which contain set-symbols. For these so called M-interpretations, soundness-resp. characterization theorems can be proved simultaneously (Theorem 2.51. Special translations of set symbols and of the formula (λωϵW)A — this means, special decisions about the size of the set W; see Sections 3 and 4 — yield the corresponding results for all interpretations of HA_ω〈〉 mentioned.

The terminology of set theoretical language — we consider an extension of HA_ω〈〉 by an extensively weak fragment only, which leads to a conservative extension of HA_ω〈〉 — is of good use for studying realizing terms of different interpretations: if HA_ω<>⊢A, A^M∃υ ∀w A_M, and ⊢A_M[t_M, w] by soundness theorem for M-interpretations, there exists a simple operation which maps $\text{v}\text{̄}\text{to}\text{t}\text{̄}{}_{\mathrm{<mtext>mr</mtext>}}$, the realizing term for modified realizability. For interpretations of Heyting's arithmetic — λ-interpretation. M-interpretations and mr-interpretation — this leads to the following stability result for existence theorems: if $\text{∃λ A}\text{and}\text{t}{}_{\mathrm{^}}\text{resp.}\text{t}{}_{\mathrm{<mtext>M</mtext><mtext>M</mtext>}}$ is the term computed by λ-interpretation. resp. M-interpretation, with $\text{∃A[t}{}_{\mathrm{<mtext>M</mtext>}}\text{]}$, then — using extensional equality and ω-rule for equations — we can prove that $\text{t}{}_{\mathrm{\lambda }}\text{= t}{}_{\mathrm{<mtext>M</mtext>}}\text{= t}{}_{\mathrm{<mtext>mr</mtext>}}$ (Section 5).