[FOM] small sets?

[William.Piper@colorado.edu]

Hello all,

I'm looking for references (or really anything) related to a question I have.

Let e be an enumeration of the set of sentences in the language of set theory
and let S be a  theory in this language. Let E(S)={the set of all reals where
the i-th digit is defined as 1 if S proves e(i), 0 if S proves ~e(i), arbitrary
otherwise}.

The set E(S) can be intuitively considered the set of all completions of S
(coded as reals). Let E(S)* be the subset of E(S) containing all and only those
reals coding consistent completions of S.

Consider the case where S is a recursively enumerable theory. What kind of set
is E(S)*? By this I mean
(I) Is E(S)* meager, could E(S)*have Lebesgue measure zero, strong measure zero
or some other "smallness" property?
(II) Could it be a projective set?
(III) Is the game G_E(S)* determined? etc.

What happens if we weaken the hypothesis a little bit and just allow S to be an
incomplete theory?

Somewhat tangential to this line of questioning, consider the the following
observation:

If T is a consistent r.e. extension of S, then the set E(T)* is a subset of
E(S)*.

Is there an analogous result for the case when T is relatively interpretable in
S?

My suspicion is that results of this type may be independent of ZFC (letting
S=ZFC or some r.e. extension of ZFC) and so...I lack any sort of intuition
about this...

Some help, please?

Everett