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Why are the Peano axioms axioms for natural numbers if they have non-standard models? Shouldn't axioms determine an object up to isomorphism?

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    $\begingroup$ Why do you think they should do that? $\endgroup$ – bof Oct 6 '20 at 11:43
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    $\begingroup$ @coffeemath they wouldn't be called models of the Peano axioms if they don't respect all of them, right? $\endgroup$ – Jiu Oct 6 '20 at 11:49
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    $\begingroup$ The second order Peano axioms do characterize the natural numbers up to isomorphism. The first order Peano axioms don't because they can't; there are non-standard models which are elementarily equivalent to the natural numbers, meaning they satisfy all the same first order sentences as the natural numbers, including the first order Peano axioms. $\endgroup$ – bof Oct 6 '20 at 11:49
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    $\begingroup$ The axioms of a vector space don't characterize vector spaces up to isomorphism. The axioms of group theory don't characterize groups up to isomorphism. $\endgroup$ – Gerry Myerson Oct 6 '20 at 11:52
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    $\begingroup$ @MauroALLEGRANZA Could you write this as an answer so that I can accept it? $\endgroup$ – Jiu Oct 6 '20 at 12:36
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The original formulation of Peano-Dedekind axioms corresponds to modern Second Order Logic version.

Richard Dedekind in The Nature and Meaning of Numbers (Was sind und was sollen die Zahlen? (1888)), proved that they characterize the natural numbers up to isomorphism.

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