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lugita15@xxxxxxxxx wrote:
> It is very easy to formalize arithmetic of finite numbers, i.e.
> First-order Peano Arithmetic.
It's much easier (better by Occam's Razor) to see that Peano's Axioms
amount to the assertion that the set of natural numbers is recursively
enuerable. Do you agree to this equivalence?
If we let P(x) mean that the set expressed by wff P is r.e., then
Peano's axioms are TRUE(x) where TRUE is the universal set. This is
10-20 times shorter and simpler.
C-B
> What I'm wondering is, whether there
> might exist such a simple axiomatization of ordinal numbers in general.
> It seems to me that such a formalization ought to contain axioms
> defining 0, the various ordinal operations such as successor, addition,
> multiplication, exponentiation etc., some way of producing limit
> ordinals, and some implementation of transfinite induction. Similar to
> PA, this axiomatization should ideally only use ordinal notions, and
> not use such thing as set theory, the von Neumann definition of
> ordinals, etc.
>
> There seems to be a problem however. Although I may be wrong, it seems
> to me that the only way of producing limit ordinals is to use the
> second-order notion of sets of ordinals. I only hope that it is
> possible to produce limit ordinals without this notion.
>
> Any help would be greatly appreciated.
>
> Thank You in Advance.
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