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From: Marc Alcobé García on 27 Jan 2010 03:40
Hi,

Again, in Levi's Basic Set Theory it is read:

"The system of axioms obtained from ZF by adding to it the axiom of
choice will be denoted ZFC. The reason for this segregation of the
axiom of choice is not because the axiom is a dubious one. It is
because ZF is sufficient for many set theoretical purposes. Also,
investigations concerning statements of set theory which are
contradicted by the axiom of choice turned out to be very interesting
and to have considerable applications to the set theory ZFC and its
extensions."

Could anyone provide some examples of the usefulness of these
investigations, specially those with applications to ZFC and its
extensions?
From: Aatu Koskensilta on 27 Jan 2010 03:45
Marc Alcob� Garc�a <malcobe(a)gmail.com> writes:

> Could anyone provide some examples of the usefulness of these
> investigations, specially those with applications to ZFC and its
> extensions?

The obvious example is the axiom of determinacy, which holds "in the
clarity of an inner model" given suitable large large cardinals, and its
use in descriptive set theory.

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

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Marc Alcobé García on 27 Jan 2010 09:21
On 27 ene, 09:45, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Marc Alcobé García <malc...(a)gmail.com> writes:
>
> > Could anyone provide some examples of the usefulness of these
> > investigations, specially those with applications to ZFC and its
> > extensions?
>
> The obvious example is the axiom of determinacy, which holds "in the
> clarity of an inner model" given suitable large large cardinals, and its
> use in descriptive set theory.
>
> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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?
From: Newberry on 27 Jan 2010 21:30
On Jan 27, 12:40 am, Marc Alcobé García <malc...(a)gmail.com> wrote:
> Hi,
>
> Again, in Levi's Basic Set Theory it is read:
>
> "The system of axioms obtained from ZF by adding to it the axiom of
> choice will be denoted ZFC. The reason for this segregation of the
> axiom of choice is not because the axiom is a dubious one. It is
> because ZF is sufficient for many set theoretical purposes. Also,
> investigations concerning statements of set theory which are
> contradicted by the axiom of choice turned out to be very interesting
> and to have considerable applications to the set theory ZFC and its
> extensions."
>
> Could anyone provide some examples of the usefulness of these
> investigations, specially those with applications to ZFC and its
> extensions?

Neither the axiom of choice nor the statements that contradict it have
any usefuness. The pragmatic test eliminates the transfinite
mathematics.
From: Newberry on 27 Jan 2010 21:31
On Jan 27, 12:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Marc Alcob Garc a <malc...(a)gmail.com> writes:
>
> > Could anyone provide some examples of the usefulness of these
> > investigations, specially those with applications to ZFC and its
> > extensions?
>
> The obvious example is the axiom of determinacy, which holds "in the
> clarity of an inner model" given suitable large large cardinals, and its
> use in descriptive set theory.

Could you expand on the "usefulness" of this axiom?
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