From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: Mathematics trivium
Date: 17 Dec 1999 04:58:44 GMT
Newsgroups: sci.math
Keywords: big numbers in proofs (Friedman -- strings in alphabets)

In article <838305$sqn$1@newsflash.concordia.ca>,
mckay@cs.concordia.ca (MCKAY john) writes:
|What is the exact size of the largest finite set studied in mathematics?

Part of what is tough is that so many of the things which generate
large numbers are parametrized, and it's harder to know in which cases
a specific case has been taken up for study.

Harvey Friedman has an example of a large number which he's "sprung"
on various people, and maybe it would almost count. Given a finite
alphabet, consider the length of the longest string a1,...,aN of
letters for which for no i<j does a_i,...,a_2i appear as a subsequence
of a_j,...,a_2j. With just one letter "x" in the alphabet, the longest
is of course xxx. With two letters x and y, the longest length is 11.
Then he asks the victim to guess the maximum for three letters. Then
(once they guess way low) he tells them a "weak" lower bound on it:
2's in a stack of exponents M high, where M is given by a stack of
exponents a trillion high. Presumably the proof involves construction
of such a sequence, which as a function is represented as a set of
ordered pairs of cardinality equal to the length of the string (and
the actual lower bound is bigger than the above).

He's got some other similar stuff which at least involves quite larger
numbers, although I would have to go look in more detail to see
whether and where actual finite sets are individually studied.

Keith Ramsay