[FOM] news?! yet again

Harvey Friedman friedman at math.ohio-state.edu
Tue Oct 28 12:59:06 EST 2003

```This is modification of the previous message. I correct some silly errors in
exponents, and make minor rearrangements.

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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.

Relation means "binary relation".

PROPOSITION. Let k,p >= 1 and R be a strictly dominating order invariant
relation on [1,2^p]k. There exists A containedin [1,2^p]k, such that every
k^2 tuple from [1,2^p] is order equivalent to a k tuple from A U. R[A],
relative to 1,2,4,...,2^p, in which 2^2^2^8k - 1 is not a coordinate.

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

A further simplification is to use qk instead of k^2, 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 be a strictly dominating order invariant
relation on [1,2^p]k. There exists A containedin [1,2^p]k, such that every
element of [1,2^p]k is order equivalent to an element of A U. R[A], relative
to 1,2,4,...,2^p, in which 2^2^2^8k - 1 is not a coordinate.

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.
Also, as in posting #192, we can use

A delta R[Ak]

A U. R[Ak].

Also note that if we remove,

2^2^2^8k - 1 does not appear

in these statements, then they become easily provable.

Harvey Friedman

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