[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.
************************************************************
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]
instead of
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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