[FOM] One Dimension/More
Harvey Friedman
friedman at math.ohio-state.edu
Mon May 26 08:46:05 EDT 2003
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.
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