[FOM] Proving PD?

[Harvey Friedman]

We continue to use the general method of Borel transfer to prove? PD.
The method was introduced in FOM postings Refuting CH?, Refuting
CH?/2.

The method starts with a very simple class of statements involving
Borel functions, and postulates that every true sentence in that class
remains true when the Borel functions are enlarged to all functions or
other larger classes.

Here are the classes of statements. K is the usual Cantor space.

1) (for all Borel S containedin K^2)(there exists continuous f:K into K)(phi).

2) (for all projective S containedin K^2)(there exists continuous f:K
into K)(phi).

Here phi is a sentences in the language of the structure (K,S,f),
where S is a binary relation symbol and f is a unary function symbol.

CONJECTURE. There is an intelligible decision procedure for 1) that
provably works in ZFC.

PROPOSITION. For every true sentence of form 1), the corresponding
sentence of form 2) is also true.

CONJECtuRE. The Proposition is provably equivalent to PD over ZFC.

THEOREM. The Proposiiton implies PD over ZFC. In fact, this is the
case where phi is an A sentence.

We can also look at

3) (for all S containedin K^2)(there exists continuous f:K into K)(phi).

Now this violates choice. We know that transfer from 1 to 3 implies AD
over ZF. We conjecture that it is equivalent over ZF.

Harvey Friedman