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1. Introduction
This is supposed to be a primer inspired by a piece by Hilbert Levitz. The theory of transfinite ordinals is a part of set theory. While the concept is tied up with the completed infinite and high cardinalities, we’ll emphasize more constructive aspects of the theory. There have been applicatons of constructive aspects of the theory. There have been applications of constructive treatments of ordinals to recursive function theory, and proof theory. Pretty much everything we need to know about ordinals follows from the following three properties of the class of all ordinals O.
2. Basic Properties of the System of Ordinal Numbers.
when is a limit ordinal.
when is limit ordinal.
when is a limit ordinal. It turns out that:
3. Cantor Normal Form
The theorem of ordinary number theory that justifies writing any non-zero number to a number base, like base 10 or base 2, applies to transfinite ordinals as well.
Theorem 1 Any non-zero ordinal can be written uniquely as a polynomial to any base greater than 1 with descending exponents and coefficients less than the base. The coefficients are written to the right of the base. Such a representation is called a Cantor normal form.
It’s common to use as the base the number , in which case the coefficients are natural numbers, thus a typical normal formlooks like:
where determine initial segments closed under addition, and numbers of the form
determine initial segments of the ordinals closed under multiplication.
4. Definition of the ordinal
then Another way to say that a number is a solution of
is to say that it’s a fixed point of the function
.
Next time we will consider computation with Ordinals on Real Computers. ;
2012-08-10 03:27:42
Source: http://peadarcoyle.wordpress.com/2012/08/09/transfinite-induction/