Some Problems in Logic: Applications of Kripke's Notion of Fulfilment

TL;DR

This paper explores Kripke's notion of fulfilment, providing new results in proof and model theory, including definable subrings, semi-representation of r.e. sets, and extensions of classical theorems, with applications to models of arithmetic and logic.

Contribution

It introduces novel results such as definable subrings with specific properties, semi-representation techniques, and extended theorems, advancing the understanding of Kripke's fulfilment in logic.

Findings

Existence of a definable subring of primitive recursive functions with specific ultrafilter properties

Feasible semi-representation of r.e. sets in theories

Complete characterization of certain sets related to conservative extensions

Abstract

This is a study of S. Kripke's notion of fulfilment. Motivated by Paris-Harrington statement, Kripke was looking for a proof of G\"odel's Incompleteness Theorem which was model-theoretic, natural (without self-reference), and easy. Fulfilment gives a versatile tool for both Proof and Model Theory. We begin with short proofs to a number of classical results. With two new results: there is an easily definable subring $R$ of the primitive recursive functions such that for any non-principal ultrafilter $D$ on $\omega$, $R/D$ is a recursively saturated model of Peano arithmetic; and for any r.e. theory $\textsf{T}$ and for any given r.e. set, we can feasibly find a $\Sigma_1^0$ formula which semi-represents it in $\textsf{T}$. We then give a version of Herbrand's Theorem, and of the Hilbert-Ackermann method of proving consistency, answering a problem of D. Guaspari:\[\{\ulcorner\phi\urcorner\in\Pi^0_k:\phi\text{ is }\Sigma^0_k\text{-conservative over $\textsf{PA}$}\}\]is a complete $\Pi^0_2$ set. We extend H. Friedman's method for results of such as $\Sigma^1_2\textsf{-AC}$ is $\Pi^1_3$-conservative over $(\Pi^1_1\textsf{-CA})_{<\varepsilon_0}\upharpoonright$, along with uniform versions\[\forall\alpha<\varepsilon_0\;(\Pi^1_1\textsf{-CA})_{\alpha}\upharpoonright\,\vdash\textsf{RFN}_{\Pi^1_3}({\Sigma^1_2}\textsf{-AC})\] Then there are some model-theoretic applications, starting with non-$\omega$-models. We extend the theorem of D. Scott involving Weak K\"onig's Lemma, and describe the order types of elementary initial segments of recursively saturated models. For $\omega$-models, we extend Friedman's theorem on minimal models of analysis, and develop indicators for countable fragments of $\mathcal L_{\infty\omega}$, with some representability results in $\omega$-logic. We close with an exposition of the Paris-Harrington statement.