[FOM] One Dimension!

[Harvey Friedman]

Here [0,infinity) is the set of all nonnegative reals. For sets of 
reals X,Y, we write X - Y = {x - y: x in X and y in Y}.

THEOREM 1. Let A,B containedin [0,infinity) be Borel. A contains or 
is disjoint from a closed C containedin [0,infinity) such that B 
containedin C - B.

It is necessary and sufficient to use all countably transfinite 
iterations of the power set operation to prove the Theorem. This 
still holds if we require that B be compact.

For projective sets A, this is equivalent to PD = projective determinacy.

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If we are more combinatorial, instead of analytic, then things are 
especially simple. Let T be the usual tree of all finite sequences of 
0's and 1's. The infinite paths through T form the usual Cantor space 
K.

THEOREM 2. Every Borel subset of K contains or is disjoint from the 
paths of a perfect subtree of T, where the set of all levels at which 
every node splits forms an infinite arithmetic progression.

It is necessary and sufficient to use all countably transfinite 
iterations of the power set operation to prove the Theorem. This 
still holds if we require that B be compact.

For projective sets A, this is equivalent to PD = projective determinacy.