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Gödel showed that a consistent, recursively enumerable axiomatic system capable of producing the standard model of natural numbers will have statements in which
by syops 6y ago
Gödel showed that a consistent, recursively enumerable axiomatic system capable of producing the standard model of natural numbers will have statements in which that statement is true in some models but not in other models.
- ProfHewitt 6y agoIn the Dedekind axiomatization of Natural Numbers, there are no standard models :-)
- syops 6y agoI was using the first order Peano axioms in my comment. I'm not familiar with the Dedekind axiomatization. Didn't even know such a thing existed but the statement I made holds I believe. The standard model under the first order Peano Axioms exists and I'm assuming it is also a model of the Dedekind axiomatization. The incompleteness theorems thus do apply to the Dedekind axiomatization, right?
- ProfHewitt 6y agoThere is just one model of Dedekind's axioms up to a unique isomorphism, which is indeed the standard model. However, Gödel's incompleteness proofs do not work for Dedekind's theory because there are uncountable axiom instances. (Dedekind's theory is nevertheless effective because proof checking is algorithmically decidable.) Consequently, the first incompleteness result (inferentially undecidable) has been proved by other means. See the following for a proof: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 The above article explains that Gödel's second incompleteness result is false because the theory can prove its own consistency.