FOM: New Classifications I
Harvey Friedman
friedman at math.ohio-state.edu
Wed Feb 20 11:01:53 EST 2002
I think I have broken the ice with regard to new clasifications. Here is
the weakest one, and I am now working on some stronger ones.
Recall from my posting #124:
PROPOSITION 1. For all multivariate f,g from N into N of quadratic
growth, there exist infinite sets A,B,C containedin N obeying
A U. fA containedin C U. gB
A U. fB containedin C U. gC.
This corresponds to Mahlo cardinals of finite order.
Notice that these are two "disjoint union inclusions" among
A,B,C,fA,fB,fC,gA,gB,gC.
I would like to figure out which sets of disjoint union inclusions can be
put there in order for the statement to be true.
This is still too hard. But I think that I can do some weaker things that I
like. Here is one that I have been able to do.
Call a dijoint union inlusion "forward" if and only if
*all capital letters on the left side stricly precede all capital letters
on the right side*.
This clearly holds of the two displayed disjoint union inclusions above.
CLASSIFICATION. For all multivariate f,g from N into N of quadratic growth,
there exist infinite sets A containedin B containedin C containedin N
obeying any given set of FORWARD disjoint union inclusions.
The statements are either provable in ACA or are provably equivalent to the
1-consistency of Mahlo cardinals of finite order over ACA.
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