Banach-Tarski paradox: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox.

Which rests on the grossly-unphysical "Axiom of Choice". That doesn't disprove cousin_it's point. Though it's still an interesting link, if that's why you're posting it.

Almost all people use mathematics where "Axiom of Choice" is set to be true. It is grossly-unusual to consider systems without that axiom.

If you are not a mathematician chances are you've never encountered a system based on axioms without "Axiom of Choice" (for systems where it is applicable at all).

I agree it's "grossly-unphysical", but you can do hardly any math without it. Now try to do physics (or computer science) without math.

A view which Gödel, an ardent Platonist, would have no doubt agreed with.

Do they, though? Is there infinity in reality?

Infinity is not a number.

Infinity is not a natural number (as is in Peano arithmetic) but in many cases it can be treated as an ordinary number e.g., for some problems on a complex plane.

You are in a good company thinking that infinity cannot be ever treated as a number http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n...

It is. Just depends which field. Surreals, superreals, hyperreals... People have been taking the reals and adding infinity all the time.

Also, there are the cardinal numbers, for which the interesting ones consist entirely of infinity.

I know, but there are more numbers than real things, or at least maybe they do. Perhaps numbers fail only in extremely large regions that we don't perceive in everyday life. (Unlikely I know, just nitpicking really).

Reality is a scary word.

Are there negative numbers? Zero? Imaginary unit?

There were times when people even didn't know a concept of zero.

A concept of infinity is really useful and it is used almost everywhere.