Epistemic systems and Flagg and Friedman's translation

Abstract

In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of the faithfulness of Gödel (or Rasiowa-Sikorski) translation $(\cdot {)}^{\square }$ of Heyting arithmetic $\mathbf{HA}$ to Shapiro's epistemic arithmetic $\mathbf{EA}$. In §2, we shall prove the faithfulness of $(\cdot {)}^{\square }$ 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 §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 §4, we shall show that epistemic Markov's rule $\mathsf{EMR}$ in $\mathbf{EA}$ is proved via $\mathbf{HA}$. So $\mathbf{EA}$ $⊢\mathsf{EMR}$ and $\mathbf{HA}$ $⊢\mathsf{MR}$ are equivalent. In §5, we shall give some relations among the translations treated in the previous sections. In §6, we shall give an alternative proof of Glivenko's theorem. In §7, we shall propose several(modal-)epistemic versions of Markov's rule for Horsten's modal-epistemic arithmetic $\mathbf{MEA}$. And, as in §4, we shall study some meta-implications among those versions of Markov's rules in $\mathbf{MEA}$ and one in $\mathbf{HA}$. 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 $\mathbf{HA}$ which has $\mathsf{DP}$ also has $\mathsf{NPE}$\rm . In §8, we shall give a proof of our \it Fundamental Conjecture \rm $\mathsf{FC}$ proposed in Inoué \cite{ino90a} as follows: $\mathsf{FC}:\mathsf{FS}⟹\mathsf{F}.$ This is a new type of proofs. In §9, I shall give discussions.

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