That such sentences are not true in all models follows from the
completeness theorem. If a sentence is not provable from certain sets
of axioms, then the sentence is not valid.
Obviously, by 'not provable from certain sets of axioms', I mean not
provable from certain sets of non-logical axioms.
Re: Godels Incompleteness and Nonmonotonic Logic ... infinite sets of axioms and derivations from them. ...Finite derivations, yes. ... then it's a logical consequence of a finite subset ... nothing to do with the completeness theorem.... (sci.logic)
Re: Do we really nedd to have models for a theory? ... we would never have LEARNED this completeness theorem.... >> One reason we care about models of a theory is that often it's ... >BEFORE we had axioms for Peano Arithmetic. ... People usually use ZFC but even that was obviously ... (sci.logic)
Re: FOL & completeness ... The completeness theorem is that if a formula is satisfied by ALL ...iff,... said axioms.... usually mentioned as itself a clause in the incompleteness theorem.... (sci.logic)
Re: Are Hilberts Axioms independent? ... |One feature of Hilbert's book is that the axioms are not first order ... the completeness theorem for first order logic doesn't apply ... every structure satisfying all of the B_i also satisfies A. Trivially, ... |one is entitled to have a model of non-Euclidean geometry,... (sci.math)
Re: Godels Incompleteness and Nonmonotonic Logic ... Gödel's completeness theorem does apply to infinite ... sets of axioms and derivations from them. ... either A is in the deductive closure or the negation of A is. ... (comp.lang.prolog)
