Re: Lebesgue but not Borel
Abhijit Dasgupta
in ZF without choice that there is any non-Borel set.
There are models of ZF in which the first uncountable
cardinal is singular, and in fact in which every set
of reals is Borel. Starting with a model of V=L,
one can build a generic extension in which every set
reals is a countable union of countable sets
(Feferman-Levy).
While the standard examples of complete analytic
or complete co-analytic sets (the universal analytic
set, the set of ordinal codes, the set of Borel codes,
etc) are all DEFINABLE in ZF without choice, to PROVE
that they are not Borel one needs at least the countable
axiom of choice. For example, in a model where aleph-1
is singular, the set WO of ordinal codes is Borel.
Abhijit Dasgupta
Dave L. Renfro
Your post gives no context regarding who you were
replying to (you might want to make a follow-up post
to your post saying which post you're replying to),
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."
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.
Dave L. Renfro
abhijit
> replying to (you might want to make a follow-up post
> to your post saying which post you're replying to),
You are correct - I replied to the following posts of 2003
via the Drexel math forum, but as the original thread
appears closed, it started a new thread with the same
name. Here are the original posts that I am refering to:
http://groups.google.com/group/sci.math/msg/718beb057f0958d1
http://groups.google.com/group/sci.math/msg/594f77b0ec1b560e
Above are the original messages of Edgar and Galvin
that I was replying to, as those posts seem to imply
that one can establish in ZF without choice that
there is a non-Borel set. (They probably really meant
that one can do only the construction in ZF without choice,
but a student of mine misinterpreted it as also asserting
that one can establish non-Borelness in ZF without choice,
which is not the case.)
(I will take a look at the posts that you mentioned.)
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 "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:
> ...
>
> Dave L. Renfro
>
I have replied in:
http://groups.google.com/group/sci.math/msg/1c76fa715eda2302
Abhijit Dasgupta
.
Dave L. Renfro
> I have replied in:
>
> http://groups.google.com/group/sci.math/msg/1c76fa715eda2302
Thanks for your detailed analysis! I've made a hard copy
of your post and filed it away with something I was working
on a year ago that I plan to get back to at some point. I
posted an outline here <http://tinyurl.com/8hexs>, but what
I posted excluded a lot of preliminary discussion on Borel
sets, Baire functions, and Young's classification of functions
that I have in my manuscript-in-progress. In particular, I
think a short summary of what can go wrong in the absence of
AC would fit in somewhere with these remarks (see below) that
I make in my manuscript about what can go wrong in more general
spaces. I'll probably want to remove the entire discussion
from the already lengthy footnote to incorporate the ~AC
possibility, however. The "two hierarchies" I refer to
are the F_alpha and G_alpha transfinite sequences of set
collections (Sigma_alpha and Pi_alpha, each with 0 superscripts,
in more modern notation).
I've removed the TEX $ characters in what follows.
---------------------------------
In general, these two hierarchies do not have to relate to
each other much more than what we've already observed, nor
do they even have to contain sets that consist of a single
point. \footnote{Singleton sets are closed sets in T_{1}
spaces (spaces X such that if x \neq y in X, then there exists
a neighborhood of each of these points that doesn't contain
the other point), but singleton sets are not necessarily Borel
in T_{0} spaces (spaces X such that if x \neq y in X, then there
exists a neighborhood of at least one of these points that
doesn't contain the other point). Harley/McNulty (1979) study
a separation axiom that lies between T_{0} and T_{1}, namely the
property that each singleton set is a Borel set. They give six
characterizations of this property and an example of a T_{0}
space such that no singleton set is a Borel set. %end footnote%}
However, in sufficiently nice spaces every closed set is a
G_delta set (any metric space will qualify) ... [I go on to
point out that this last property implies the two hierarchies
interleaf each other from this stage onward in the usual
manner.]
Peter W. Harley and George F. McNulty, "When is a point Borel?"
Pacific Journal of Mathematics 80 (1979), 151-157.
[MR 80e:54020; Zbl 403.54034]
http://www.emis.de/cgi-bin/MATH-item?0403.54034
---------------------------------
Dave L. Renfro