Re:Hawking Hawking
by syops on Monday April 12, 2021 @10:03AM
(#26779563)
I 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?
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Re:Hawking Hawking
by ProfHewitt on Friday April 16, 2021 @11:55AM
(#26835494)
There 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
The above article explains that Gödel's second
incompleteness result is false because the theory can prove
its own consistency.