Epistemic systems and Flagg and Friedman's translation

TL;DR

This paper introduces a modified translation method to prove faithfulness of the G"odel translation from Heyting arithmetic to epistemic arithmetic, simplifying previous proofs and exploring applications to properties like disjunction and numerical existence, along with related theorems.

Contribution

It presents a new translation that simplifies the proof of faithfulness of G"odel's translation and applies it to various properties and theorems in epistemic and intuitionistic arithmetic.

Findings

Simplified proof of faithfulness of G"odel translation

Applications to disjunction and numerical existence properties

Proof of the Fundamental Conjecture relating modal and classical theorems

Abstract

In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of the faithfulness of G\"{o}del (or Rasiowa-Sikorski) translation $(\cdot)^\Box$ of Heyting arithmetic $\bf HA$ to Shapiro's epistemic arithmetic $\bf EA$. In \S 2, we shall prove the faithfulness of $(\cdot)^\Box$ without using stability, by introducing another translation from an epistemic system to corresponding intuitionistic system which we shall call \it the modified Rasiowa-Sikorski translation\rm . That is, this introduction of the new translation simplifies the original Flagg and Friedman's proof. In \S 3, we shall give some applications of the modified one for the disjunction property ($\mathsf{DP}$) and the numerical existence property ($\mathsf{NEP}$) of Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule $\mathsf{EMR}$ in $\bf EA$ is proved via $\bf HA$. So $\bf EA$ $\vdash \mathsf{EMR}$ and $\bf HA$ $\vdash \mathsf{MR}$ are equivalent. In \S 5, we shall give some relations among the translations treated in the previous sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In \S 7, we shall propose several(modal-)epistemic versions of Markov's rule for Horsten's modal-epistemic arithmetic $\bf MEA$. And, as in \S 4, we shall study some meta-implications among those versions of Markov's rules in $\bf MEA$ and one in $\bf HA$. Friedman and Sheard gave a modal analogue $\mathsf{FS}$ (i.e. Theorem in \cite{fs}) of Friedman's theorem $\mathsf{F}$ (i.e. Theorem 1 in \cite {friedman}): \it Any recursively enumerable extension of $\bf HA$ which has $\mathsf{DP}$ also has $\mathsf{NPE}$\rm . In \S 8, we shall give a proof of our \it Fundamental Conjecture \rm $\mathsf{FC}$ proposed in Inou\'{e} \cite{ino90a} as follows: $\mathsf{FC}: \enspace \mathsf{FS} \enspace \Longrightarrow \enspace \mathsf{F}.$ This is a new type of proofs. In \S 9, I shall give discussions.