FOM: New Classifications I

[Harvey Friedman]

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.