Author:
OMODEO EUGENIO G.,POLICRITI ALBERTO,TOMESCU ALEXANDRU I.
Abstract
This paper considers ∃*∀* prenex sentences of pure first-order predicate calculus with equality. This is the set of formulas which Ramsey's treated in a famous article of 1930. We demonstrate that the satisfiability problem and the problem of existence of arbitrarily large models for these formulas can be reduced to the satisfiability problem for ∃*∀* prenex sentences of Set Theory (in the relators ∈, =).We present two satisfiability-preserving (in a broad sense) translations Φ ↦ $\dot{\Phi}$ and Φ ↦ Φσ of ∃*∀* sentences from pure logic to well-founded Set Theory, so that if $\dot{\Phi}$ is satisfiable (in the domain of Set Theory) then so is Φ, and if Φσ is satisfiable (again, in the domain of Set Theory) then Φ can be satisfied in arbitrarily large finite structures of pure logic.It turns out that |$\dot{\Phi}$| = $\mathcal{O}$(|Φ|) and |Φσ| = $\mathcal{O}$(|Φ|2).Our main result makes use of the fact that ∃*∀* sentences, even though constituting a decidable fragment of Set Theory, offer ways to describe infinite sets. Such a possibility is exploited to glue together infinitely many models of increasing cardinalities of a given ∃*∀* logical formula, within a single pair of infinite sets.
Publisher
Cambridge University Press (CUP)
Subject
Computer Science Applications,Mathematics (miscellaneous)
Cited by
8 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献