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From: Frederick Williams on 28 Jan 2010 13:28
Marc Alcob� Garc�a wrote:
>
> > If all subsets of R are measurable then AC is false.
>
> Do you refer to Lebesgue measurable subsets? The existence of non-
> Lebesgue measurable sets can be proved directly from ZFC applying
> Zorn's Lemma.

Yes, sorry, Lebesgue measurable.

--
Mathematics is a part of physics.
Physics is an experimental science, a part of natural science.
Mathematics is the part of physics where experiments are cheap.
(V.I. Arnold)
From: Aatu Koskensilta on 1 Feb 2010 10:22
Marc Alcob� Garc�a <malcobe(a)gmail.com> writes:

> Could you extend a bit more on this? I mean, what does the axiom of
> determinacy reveal about the structure of sets as described by ZFC and
> its extensions?

Determinacy (together with dependent choice) provides a nice and rich
theory of sets of real numbers. When determinacy was first proposed it
was conjectured that although it doesn't hold in the actual universe of
sets there should be some restricted universe of sets in which it does
hold. And indeed, we now know that if there are infinitely many Woodin
cardinals determinacy (and dependent choice) holds in the inner model
L(R) of set theory, of great importance in descriptive set theory. (The
various classes of sets of reals studied in descriptive set theory all
live in L(R)).

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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