In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of the faithfulness of G\"{o}del 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 propose a modified version of \it Fundamental Conjecture \rm $\mathsf{FC}$ ($\mathsf{FS} \Longrightarrow \mathsf{F}$) proposed by the author as $\Delta_0$-Fundamental Conjecture. In \S 9, I shall give some discussions and my philosophy.

翻译：1986年，Flagg和Friedman \cite{ff}给出了Heyting算术$\bf HA$到Shapiro认识论算术$\bf EA$的Gödel翻译$(\cdot)^\Box$忠实性的优雅替代证明。在第2节中，我们将通过引入一种从认识论系统到相应直觉主义系统的新翻译（称为修正的Rasiowa-Sikorski翻译），在不使用稳定性概念的情况下证明$(\cdot)^\Box$的忠实性。也就是说，这个新翻译的引入简化了Flagg和Friedman的原始证明。在第3节中，我们将给出修正翻译在Heyting算术的析取性质（$\mathsf{DP}$）和数值存在性质（$\mathsf{NEP}$）方面的一些应用。在第4节中，我们将证明$\bf EA$中的认识论马尔可夫规则$\mathsf{EMR}$可以通过$\bf HA$得到证明，从而$\bf EA$ $\vdash \mathsf{EMR}$与$\bf HA$ $\vdash \mathsf{MR}$是等价的。在第5节中，我们将给出前几节中讨论的翻译之间的一些关系。在第6节中，我们将给出Glivenko定理的另一种证明。在第7节中，我们将为Horsten的模态-认识论算术$\bf MEA$提出马尔可夫规则的若干（模态）认识论版本。与第4节类似，我们将研究$\bf MEA$中这些马尔可夫规则版本与$\bf HA$中马尔可夫规则之间的元蕴含关系。Friedman和Sheard给出了Friedman定理$\mathsf{F}$（即\cite{friedman}中的定理1）的模态类比$\mathsf{FS}$（即\cite{fs}中的定理）：\it 任何具有$\mathsf{DP}$的$\bf HA$的递归可枚举扩展也具有$\mathsf{NPE}$\rm。在第8节中，我们将提出作者提出的\it基本猜想\rm $\mathsf{FC}$（$\mathsf{FS} \Longrightarrow \mathsf{F}$）的修正版本，称为$\Delta_0$-基本猜想。在第9节中，我将给出一些讨论和我的哲学观点。