|
|
Charlie-Boo wrote:
> 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?
>
I do not care which axiomatization is most efficient. I am only using
Peano's axioms as an example of what I want. I do not care if my
requested axiomatization of ordinal arithmetic is inefficient. All I
care is that it satisfies the properties I want.
> 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.
|
|