[FOM] Big numbers - confusion
Harvey Friedman
friedman at math.ohio-state.edu
Tue Mar 28 23:05:20 EST 2006
The confusion about big numbers apparently started with the posting
http://www.cs.nyu.edu/pipermail/fom/2006-March/010271.html
which didn't directly describe the number in question, but only quoted two
Theorems of mine that IMPLICITLY described the number in question:
> THEOREM 1. There exists n >= 1 such that the following holds. Let
> T_1,...,T_n be finite trees with vertices labeled from {1,...,6}, where each
> T_i has at most i vertices. There exists 1 <= i < j <= n such that T_i is
> inf preserving and label preserving embeddable into T_j.
>
> THEOREM 2. Theorem 1 can be proved in strictly finite mathematics. However,
> any such proof in ACA_0 + Pi12-BI must use at least 2^[1000] symbols.
>
> Here 2^[1000] is an exponential stack of 2's of height 1000.
The number in question was incorrectly taken to be 2^[1000] by some
subscribers.
The number in question is obviously the number discussed in my posting
http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html
Theorem ##. I write it as TREE[6], although I actually treat the smaller
number TREE[3]. See the remarks at the end about the comparisons with my
previous n(3), and Graham's number.
Now I am quite certain that the author of the original posting that started
the confusion knew exactly what he meant by "my number". As for the authors
of
http://www.cs.nyu.edu/pipermail/fom/2006-March/010275.html
http://www.cs.nyu.edu/pipermail/fom/2006-March/010277.html
I cannot say the same. Now there were a lot of numbers explicitly mentioned
in Theorems 1 and 2 above:
1
2
6
1000
2^[1000].
At least it was nice to see that their eyes focused on 2^[1000] rather than
the others!
Harvey Friedman
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