- #26

Hurkyl

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Hausdorffwhen i meant construction of natural numbers i meant you define 0=empty set 1={ES} etc, and you define what it means to be an inductive set. and then you need the axiom of infinity to deduce that there exists an inductive set (yes i can see hausdorff's point better to keep the numbers undefined (-:)... and so on.\now when i rethink it, it really depends in the context, if you were discussing it in any discpline besides logic and set theory you shouldnt have this construction but then agian hausdorff's comment is in a set theory textbook, you should expect this kind of constrcution would you not?

**was not**saying that you should not make this construction -- he was not saying that you should not prove that the finite ordinals are a model of the natural numbers.

Hausdorff

**was**saying that you shouldn't read too much into this construction -- you should not think of this construction as saying what the natural numbers "are".

When you prove that the finite ordinals under addition satisfy the monoid axioms, do you think "Aha! The monoid axioms were really just defining the finite ordinals!"? I assume not -- so when you prove that the finite ordinals under the successor operation satisfy Peano's axioms, why would you think "Aha! Peano's axioms were really just defining the finite ordinals!"?

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