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Assume consistency. Take the collections of all true statements about the standard model of the natural numbers and make these statements your axioms. Now ever
by syops 6y ago
Assume consistency. Take the collections of all true statements about the standard model of the natural numbers and make these statements your axioms. Now every true statement about the standard model of the natural numbers is provable. It’s a useless axiomatic system since there is no procedure for knowing if a statement is an axiom of not (that works for every statement). The second order Peano axioms are categorical and assuming consistency every true statement is ‘provable’.
- ProfHewitt 6y agoHigher-order Dedekind/Peano axioms are categorical in the sense that they characterize Natural Numbers up to a unique isomorphism. However, there is still a proposition about Natural Numbers that true but unprovable using the Dedekind/Peano axioms. Further information can be found here: "Physical Indeterminacy in Digital Computation" https://papers.ssrn.com/abstract=3459566 https://papers.ssrn.com/abstract=3459566
- syops 6y agoSo the last sentence I made is false. The rest is correct, right?
- ProfHewitt 6y agoYes, the last statement that you made is indeed false, that is, it is not the case that every true proposition is provable in Dededkind's theory.