Wednesday, September 24, 2008

Finite implications of transfinite math

Harvey Friedman has a dream in which the large cardinal axioms of set theory are used all the time, not just by set theorists, but also by combinatorialists (or "finite-set theorists" as I like to think of them). I share his dream. How do we get there? Here's Friedman's strategy:
The general strategy for using large cardinals in the integers is as follows. We start with a discrete (or finite) structure X obeying certain hypotheses H. We wish to prove that a certain kind of finite configuration occurs in X, assuming that H holds. We define a suitable concept of completion in the context of arbitrary linearly ordered sets. We verify that if X has a completion with the desired kind of finite configuration, then X already has the desired kind of finite configuration. We then show, using hypotheses H, that X has completions of every well-ordered type. We now use the existence of a suitably large cardinal Lambda. Using large cardinal combinatorics, we show that in any completion of order type Lambda, the desired kind of finite configuration exists. Hence the desired kind of finite configuration already exists in X.
If you follow the link, you see in detail a particular instance of this strategy. The question is, why hasn't this been done more often? Is it a sociological problem or a mathematical problem?

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