# [FOM] news?!

Harvey Friedman friedman at math.ohio-state.edu
Tue Oct 28 12:25:51 EST 2003

```It looks like simplifications are coming in on posting #192. I need to see
how far I can get, but here is what is happening.

PROPOSITION. Let k,p >= 1 and R containedin [1,2^p]2k be strictly dominating
and order invariant. There exists A containedin [1,2^p]k, without 2^2^2^8k -
1, such that every k^k tuple from [1,2^p] is order equivalent to a k tuple
from A U. R[Ak], relative to 1,2,4,...,2^p.

This Proposition appears to be equivalent to the consistency of Mahlo
cardinals of finite order.

A further simplification is to use k^q instead of k^k, and q instead of k,
where q is TINY.

The case q = 3 seems very likely, q = 2 likely, and q = 1 reasonable. I mean
the reversals.

So the current target is

PROPOSITION'. Let k,p >= 1 and R containedin [1,2^p]2k be strictly
dominating and order invariant. There exists A containedin [1,2^p]k, without
2^2^2^8k - 1, such that every k tuple from [1,2^p] is order equivalent to an
element of A U. R[Ak], relative to 1,2,4,...,2^p.

With a little bit of luck, I should see how to reverse Proposition'.

(As in posting #192, 2^2^2^8k - 1 is meant to be a silly but safe number).

Harvey Friedman

```