FOM: 125:Disjoint Unions/First Classifications

[Harvey Friedman]

This is a followup to posting #125, 6:19AM  3/1/02.

We have made another classification, this time without growth condition. We
had fully expected that without the grwoth condition, the classification
would NOT turn up any statements independent of ZFC, and our expectation
was confirmed.

Thus in posting #125, we discussed the classification of all statements of
the form:

For all multivariate f,g from N into N of quadratic growth, there exist
infinite sets A,B,C containedin N such that

X U. fY containedin V U. gW
X' U. fY' containedin V' U. gW'

where X,Y,V,W,X',Y',V',W' are among the letters A,B,C.

We now discuss the classification of all statements of the form:

For all multivariate f,g from N into N, there exist infinite sets A,B,C
containedin N such that

X U. fY containedin V U. gW
X' U. fY' containedin V' U. gW'

where X,Y,V,W,X',Y',V',W' are among the letters A,B,C.

THEOREM. Every such statement is provable or refutable in RCA0.

This is in obvious contrast to the previous classification in #125, where
there are exactly six instances which are independent of ZFC (exactly 1 up
to permutation of the three letters A,B,C).