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

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 $\omega$, $R\mathrm{/}D$ is a recursively saturated model of Peano arithmetic; and for any r.e. theory $\mathsf{\text{T}}$ and for any given r.e. set, we can feasibly find a ${\mathrm{\Sigma }}_1^0$ formula which semi-represents it in $\mathsf{\text{T}}$. We then give a version of Herbrand’s Theorem, and of the Hilbert-Ackermann method of proving consistency, answering a problem of D. Guaspari:$$\{\varphi \in {\mathrm{\Pi }}_k^0:\varphi \text{ is }{\mathrm{\Sigma }}_k^0\text{-conservative over }\mathsf{\text{PA}}\}$$is a complete ${\mathrm{\Pi }}_2^0$ set. We extend H. Friedman’s method for results of such as ${\mathrm{\Sigma }}_2^1\mathsf{\text{-AC}}$ is ${\mathrm{\Pi }}_3^1$-conservative over $({\mathrm{\Pi }}_1^1\mathsf{\text{-CA}}{)}_{<{\epsilon }_0}\upharpoonright$, along with uniform versions$$\mathrm{\forall }\alpha <{\epsilon }_0({\mathrm{\Pi }}_1^1\mathsf{\text{-CA}}{)}_{\alpha }\upharpoonright ⊢{\mathsf{\text{RFN}}}_{{\mathrm{\Pi }}_3^1}({\mathrm{\Sigma }}_2^1\mathsf{\text{-AC}})$$ 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 ${\mathcal{L}}_{\mathrm{\infty }\omega }$, with some representability results in $\omega$-logic. We close with an exposition of the Paris-Harrington statement.