Re: Axiomatization of Ordinal Arithmetic

It is very easy to formalize arithmetic of finite numbers, i.e.
First-order Peano Arithmetic. 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.

If I remember right, Tarski's book "Ordinal Algebras" does something
like that.

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Relevant Pages

  • Re: Axiomatization of Ordinal Arithmetic
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  • Re: is there a set of all ordinals
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    (sci.logic)
  • Re: Axiomatization of Ordinal Arithmetic
    ... It's much easier to see that Peano's Axioms ... I do not care which axiomatization is most efficient. ... ordinals, and some implementation of transfinite induction. ... to me that the only way of producing limit ordinals is to use the ...
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