FOM: ReplyToMarker
Harvey Friedman
friedman at math.ohio-state.edu
Mon Nov 10 11:34:54 EST 1997
Marker writes:
>A) Let E be a Borel subset of the unit square in the plane, which is symmetric
>about the line y = x. Then E contains or is disjoint from the graph of a
>Borel measurable function.
>
>Harvey showed that A) is true but not provable in (or well beyond) second
>order arithmetic.
I actually showed that this is provable using uncountably many uncountable
cardinals (via Borel determinacy as you indicated), but not with any
countable number of uncountable cardinals. In fact, I proved that it is
outright equivalent to for all reals x and countable ordinals alpha, there
is an omega-model of V(alpha) containing x. This is done in the paper
On the Necessary Use of Abstract Set Theory, Advances in Math., vol. 41,
No. 3, 1981, pp. 209-280.
>B) Let E be an analytic subset of the unit square in the plane, which is
>symmetric
>about the line y = x. Then E contains or is disjoint from the graph of a
>Borel measurable function. (Unlike my previous post, here by "analytic" I
>mean the
>projection of a Borel set in R^3.)
>
>B) is again an easy consequence of analytic determinacy. By theorems of
>Martin and
>Harrington (another f.o.m. highlight) we know that analytic determinacy is
>equivalent to the
>existence of sharps.
I didn't consider this question there since I was concerned with the Borel
world and absoluteness - i.e., the vitally important concreteness one gains
by staying within the Borel world. But if my memory serves me right, this
is true. In fact, if I remember correctly, given any reasonable class K of
subsets of reals, determinacy is equivalent to "every symmetric subset of
IxI lying in K contains or is disjoint from the graph of a Borel function."
Again if my memory is correct, this equivalence gets going above Borel, and
is first due to Kunen. Hence the cc of this posting.
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