[FOM] One Dimension/More

[Harvey Friedman]

THEOREM 1.1. There is a continuous function H:[0,1] into [0,1] such 
that the following holds. Every Borel subset of [0,1] contains or is 
disjoint from a closed set whose image under H is [0,1].

Using well known effectivity notions, we can simplify even further.

THEOREM 1.2. Every Borel subset of [0,1] contains or is disjoint from 
a closed set whose image under some recursively continuous function 
from [0,1] into [0,1], is [0,1].

Or the more liberal notion of arithmetical.

THEOREM 1.3. Every Borel subset of [0,1] contains or is disjoint from 
a closed set whose image under some arithmetical function from [0,1] 
into [0,1], is [0,1].

It is necessary and sufficient to use uncountably many iterations of 
the power set operation to prove any of Theorems 1.1 - 1.3. If we 
replace Borel by "projective" then we get PD. Etc.

There are many interesting technical strengthenings of Theorem 1.1 
that have precisely the same metamathematical status. Some are 
combinatorial intriguing.

Here's one that is at least implicitly well known.

Let K be the Cantor space of infinite bit sequences, K. We count the 
positions from 1 on.

THEOREM 1.5. Every Borel subset of K either contains a perfect set 
whose elements have arbitrary bits at the even positions, or is 
disjoint from a perfect set whose elements have arbitrary bits at the 
odd positions.