Monday, March 29, 2004

There, that project is done. Well mostly done. I finally finished the first draft of A Tale of Two Paradoxes. I suppose that I will have to someday look at it again and correct all the spelling and grammar mistakes. I wish I could afford an editor. I find I can do a passable editing job if I leave and article alone for a few months, then revisit it.

Anyway, this is the latest effort to address the paradoxes of transfinite theory. It has a few modest improvements over the previous efforts. It strongly emphasize that the diagonal method is simply a form of the liar's paradox. The two paradoxes are Galileo's paradox and the liar's paradox. Transfinite theory uses Galileo's paradox to assert that the set of rationals is the same "size" as the integers, then uses the liar's paradox to say that the real numbers are a different size. This is the denumerable/nondenumerable dichotomy. Mathematics is supposed to somehow arise from the opposition of these two terms.

The purpose of this set of articles is to address the use of paradoxes as the foundations of mathematics. It is not about the conclusions of theory, but about the foundations. So I added a few words on the difference between discrete and continuous mathematics. Discrete mathematics works only with terms of finite length. Continuous math uses terms of infinite length. The differences between these branches arise from definitions, not paradoxes.

We can create only a finite number of unique strings from a finite number of characters. Expressing a totality, like the set of real numbers, requires an infinite number things: Discrete mathematics is content to work with finite entities. Continuous mathematics must allow for the use of infinite terms. The difference between the subjects does not arise from a fundamental dichotomy created by paradoxes.

The article does not deny the existence of paradoxes. It simply says that they sould be treated as a side dish. For example, we should not use Galileo's paradox as the definition of the infinite!!!!! Anyway, I hate this work on transfinite theory. I will not let the site fester for a few months and work on things I enjoy.

No comments: