Paradoxes of the infinite

[Frederick Williams]

The discussion in 'Matheology \S 071' bought to mind something of
Kreisel's that I read years ago[1]: "... Hausdorff's sphere paradox
depends ultimately on explicitly defined denumerable disjoint sets A, B,
C such that A can be rotated both into B and C separately, and also into
B union C. This is 'geometrically' paradoxical, but nothing else but an
elaboration of aleph_0 + aleph_0 = aleph_0, and so has nothing to do
with the axiom of choice."

[1] G. Kreisel 'Comments' to Mostowski 'Recent results in set theory' in
'Philosophy of mathematics', ed. Lakatos, N-H, 1972. ('Tis on page
103.)

Now, I thought that the complement of A u B u C was countable, so at
least one of A, B, C was uncountable. I must have misunderstood
something.

[I also think that the Hahn-Banch theorem can be used in place of Choice
to prove the Banach-Tarski paradox. But that's another story.]

--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.

[William Elliot]

There are infinitely many paradoxes of the infinite for
if there were only finite many paradoxes it would be finite.

[Ross A. Finlayson]

On Jul 22, 7:20 pm, William Elliot <ma...@panix.com> wrote:
> There are infinitely many paradoxes of the infinite for
> if there were only finite many paradoxes it would be finite.

No, there's none.

In our understanding of the finite and infinite, there are some
antinomies or paradoxes (eg Cantor's, Russell's, Burali-Forti's), but
a strong platonist may well find there are none, in a real reality
with real infinities. There are resolutions of those (and not in ZF).

We're quite well past, for example, those of Zeno of Elea, of the
archer and Achilles and the tortoise, here thanks to Cauchy and
Weierstrass, as an example of the elucidation of a framework with
methods of exhustion, here that of the integral and differential
calculus.

There are infinitely many things else they would be finite and a dot.

Regards,

Ross Finlayson

[William Elliot]

On Sun, 22 Jul 2012, Ross A. Finlayson wrote:
> On Jul 22, 7:20�pm, William Elliot <ma...@panix.com> wrote:

> > There are infinitely many paradoxes of the infinite for
> > if there were only finite many paradoxes it would be finite.
>
> No, there's none.
>
> In our understanding of the finite and infinite, there are some
> antinomies or paradoxes (eg Cantor's, Russell's, Burali-Forti's), but
> a strong platonist may well find there are none, in a real reality
> with real infinities. There are resolutions of those (and not in ZF).

Infinity is too big for realists.

> We're quite well past, for example, those of Zeno of Elea, of the
> archer and Achilles and the tortoise, here thanks to Cauchy and
> Weierstrass, as an example of the elucidation of a framework with
> methods of exhustion, here that of the integral and differential
> calculus.
>
> There are infinitely many things else they would be finite and a dot.

Ad infinitum.

> Regards,
>
> Ross Finlayson
>

[Frederick Williams]

Frederick Williams wrote:

>
> [I also think that the Hahn-Banch theorem can be used in place of Choice
> to prove the Banach-Tarski paradox.

The point being that Hahn-Banach (correcting the spelling) is weaker
than Choice.