# [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^ symbols.
>
> Here 2^ is an exponential stack of 2's of height 1000.

The number in question was incorrectly taken to be 2^ 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, although I actually treat the smaller
number TREE. 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^.

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

Harvey Friedman

```