## Some Problems in Logic: Applications of Kripke’s Notion of Fulfilment

Por • 4 may, 2019 • Sección: Educacion

J.E. Quinsey

This is a study of S. Kripke’s notion of fulfilment. Motivated by Paris-Harrington statement, Kripke was looking for a proof of Gödel’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 ω, R/D is a recursively saturated model of Peano arithmetic; and for any r.e. theory T and for any given r.e. set, we can feasibly find a Σ01 formula which semi-represents it in 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:

{┌ϕ┐∈Π0k:ϕ is Σ0k-conservative over PA}

is a complete Π02 set.  We extend H. Friedman’s method for results of such as Σ12-AC is Π13-conservative over (Π11-CA)<ε0↾, along with uniform versions

∀α<ε0(Π11-CA)α↾⊢RFNΠ13(Σ12-AC)

Then there are some model-theoretic applications, starting with non-ω-models. We extend the theorem of D. Scott involving Weak König’s Lemma, and describe the order types of elementary initial segments of recursively saturated models. For ω-models, we extend Friedman’s theorem on minimal models of analysis, and develop indicators for countable fragments of L∞ω, with some representability results in ω-logic.  We close with an exposition of the Paris-Harrington statement.

arXiv:1904.10540v1 [math.LO]

Logic (math.LO)