FOM: New Classifications 2

[Harvey Friedman]

A continuation of 11:01AM  2/20/02.

We now move to classifications which don't involve an inclusion tower A
containedin B containedin C containedin N and don't involve any notion of
largeness.

CLASSIFICATION. For all multivariate f,g from N into N of quadratic growth,
there exist infinite sets A1,A2,A3 containedin N obeying any given set of
disjoint union inclusions

Ai U. fAj containedin Ap U. gAq

where i,j < p,q.

The statements in the classification are either provable in ACA or are
provably equivalent to the 1-consistency of Mahlo cardinals of finite order
over ACA.

Obviously the "master" case of these two:

A1 U. fA1 containedin A3 U. gA2
A1 U. fA2 containedin A3 U. gA3

is a particularly simple special case. Recall that this case is provably
equivalent to the 1-consistency of Mahlo cardinals of finite order over ACA.

As usual, I can use other related notions of healthy growth, and get the
same classification.

(For those just tuning in, U. is "disjoint union").