>>>"We in fact know many things that are infinite..."
No, we don't. Infinity is only a mathematical trick or, as Richard says, 'a labelled mathematical concept.'
A set is a mental concept. A number is a mental concept. But these things have no more bearing on what they represent than the letter N, or the concept "two syllables" have on a neutron. We might posit, "the set of real numbers is infinite," but what are we really saying?
Richard made the proper distinction at the onset of this discussion: "infinity does not exist, except as a labelled mathematical concept."
I think that statement hit the nail squarely on the head. Genuine infinity is a logical paradox.
I think a logical examination of infinity is perfectly in sync with Godel; while logic suggests infinity might be possible, an infinity would be impossible to prove logically.
Consider - a genuine infinity would have to be infinitely infinite. A sphere, for instance, is not infinitely infinite. It's merely boundless. Anything truly infinite would manifest itself infinitely.
Everything we contemplate occupies time, doesn't it? Can we conceive of a thing exempt from the phenomenon of time? I think not -- the fact that we would obviously have to take some time to contemplate a possibly infinite thing (p.i.t.) shows that the p.i.t. itself -- in some way -- occupies time. And since a truly infinite phenomenon would have to be infinitely infinite, this p.i.t. would have to occupy all of time, lest it be merely boundless.
Remember -- infinity means infinitely infinite.
You may assert that a thing occupies all of time, has always existed and will always exist, but proving this would be futile, as this assertion would lie beyond the capacity of logic. After all, logic cannot omit cause and effect, can it? The asserted p.i.t. could not be logically proved...
...unless we wrap a boundary around it. At which time it is no longer infinite. Etc.
Sounds silly, but I think a consideration of infinity becomes silly because the concept is mathematically paradoxical. Either we dump logic, we dump the idea of infinity, or we agree with what Godel told us -- math isn't perfect!
Which is why infinity is a logical paradox.