[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


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

The number in question is obviously the number discussed in my posting


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


I cannot say the same. Now there were a lot of numbers explicitly mentioned
in Theorems 1 and 2 above:


At least it was nice to see that their eyes focused on 2^[1000] rather than
the others!

Harvey Friedman 

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