Re: Every set is F_sigma-delta-sigma in the Feferman-Levy model?
Abhijit Dasgupta
http://groups.google.com/group/sci.math/msg/745955494f579c61
http://groups.google.com/group/sci.math/msg/72bbc723f0ed3cd7
Abhijit Dasgupta
Abhijit Dasgupta
> but I found it interesting because a few years ago
> I asked a question that maybe you can add to what
> a couple of others said back then.
>
> sci.math thread "Lebesgue but not Borel", post #1
> (January 1, 2006) [your post, FYI]
> http://groups.google.com/group/sci.math/msg/2aa51af72a047c8a
>
> sci.math thread "Every set is F_sigma-delta-sigma in the
> Feferman-Levy model?", post #1 (July 30, 2001)
> http://groups.google.com/group/sci.math/msg/745955494f579c61
>
> In particular, I wonder if you can help with these questions
> from the end of my 2001 post:
>
> "My guess is that without any infinite version of AC to fall
> back on, even the notion of an F_sigma-delta-sigma set might
> have more than one interpretation. So perhaps saying that
> every subset of the reals is F_sigma-delta-sigma might be
> an oversimplification of the actual result."
(The Feferman-Levy construction is described in Jech's
book Set Theory. See either the 3rd (millenium) edition
Example 15.57 pp. 259--260, or the 2nd edition
Example IV in Ch 3 Sec 21 pp. 213--214.)
I think you are right in pointing out that there is
a chance of confusion in interpreting a class of sets
described by a term such as "F_sigma-delta-sigma",
or even a term like "F_sigma-sigma" or "F_sigma-delta".
Care needs to be taken in using these terms (which
involve multiple infinitary operations) in absence of AC.
1. The first interpretation: This is perhaps the more
natural one, as the order of the operations in the
notation clearly suggests that "F_sigma-delta"
means "countable intersection of F_sigma sets".
So here "F_sigma-delta" really stands for
"((F)_sigma)_delta", where the notation "(C)_sigma"
denotes all countable union of sets from a class C;
and similarly for "(C)_delta". Another example of
this interpretation: "F_sigma-delta-sigma" really
stands for "(((F)_sigma)_delta)_sigma)".
2. The second interpretation: This is a more uniform,
effective version, and defines smaller classes of sets.
In this interpretation, "C_sigma-delta" stands for
"C_(sigma-delta)", where the notation "C_(sigma-delta)"
is used to denote the class of all sets of the form
/\_m \/_n C_{m,n}, where C_{m,n} is an N x N array
of sets, each from the class C. (Similarly for
"C_(sigma-sigma)", etc)
Now even in ZF without choice it is true that
C_(sigma-delta) is contained in ((C)_sigma)_delta,
but the reverse inclusion may not be true unless
countable AC is used. Without countable AC,
C_(sigma-sigma) may be a proper subset of
((C)_sigma)_sigma. Etc. In general, for any term
like "F_sigma-delta-sigma", the first interpretation
gives a larger collection, and the second interpretation
defines a smaller collection. If DC is assumed,
the two interpretations will coincide.
If the second interpretation is used, all the usual
properties of the finite levels of the Borel hierarchy
(such as universal sets, non-degenerateness of levels,
etc) remain valid even without choice.
But if the first interpretation is used and
AC is not assumed, then this usual structure for the
finite levels of the Borel hierarchy is no longer
available (for example, F_sigma may then be a proper
subset of F_sigma-sigma, F_sigma-delta may not have
a universal set, F_sigma-delta-sigma may equal
G_delta-sigma-delta, etc).
When interpreting statements with respect to the
Feferman-Levy model (where even countable choice
fails), one needs to use the first interpretation.
With that interpretation, one can say the following:
(See Jech) If you start with a ground model
satisfying GCH, then in the Feferman-Levy extension
the set of reals is a countable union of countable sets
(thus even the countable AC fails badly). Also,
since every countable set is an F_sigma, so in
this extension, every set of reals is an F_sigma-sigma
(and so also F_sigma-delta-sigma) and G_delta-delta
(and so also G_delta-sigma-delta).
>
> For example, I assume something technical is going on because
> one would naively expect every subset of the reals to be
> F_sigma in the Feferman-Levy model, or perhaps as G. A. Edgar
> commented in that thread, F_sigma-sigma.
As I mentioned above, and as Edgar mentions, in the
Feferman-Levy model, every set of reals will be
F_sigma-sigma, but this does not imply (even naively)
that every set of reals is also F_sigma. Thus,
in the Feferman-Levy model, F_sigma is a proper subset
of F_sigma-sigma, while F_sigma-sigma equals the
power set of R. All this makes sense because we are
using the first interpretation.
To be sure, we certainly have, even in the Feferman-Levy
model, many sets of reals (e.g. the set of irrational
numbers, or any other dense G_delta set) which are not
F_sigma, as this fact is provable in ZF without choice.
Abhijit Dasgupta
>
> Dave L. Renfro
>