[tambourine_man]

That has always bothered me deeply.

We measure something by comparing its frontiers (where it begins and ends) to something else. If this something has none, this clearly cannot be done.

Of course, it's also clear that between 0 and 1 you have infinite real numbers. But is an infinite inside an infinite enough to provide a size hierarchy?

Maybe the quantum physics guys will prove that the universe is granular in every possible level and that everything is just an enormous pile of huge natural numbers. Then infinity and paradoxes will just be a fun thought experiment, and reality will still be pragmatically ungraspable, but profoundly boring.

[mturmon]

Don't be bothered -- this is not actually how it's done.

What you do to compare sizes of sets A and B is, construct a 1:1 function mapping everything in A to something in B and vice versa.

If such a function exists, they're the same cardinality.

It's a little long-winded, but see:

http://en.wikipedia.org/wiki/Cardinal_number

This idea, which seems obvious only in retrospect, is due to Georg Cantor (the guy with the set, and the paradox).

[tambourine_man]

Yeah, but if natural numbers are infinite, than there will always be a natural number to map to a real one, therefore they're the same cardinality.

[Chirono]

Strangely enough, this isn't actually true! Ok, so it is true that if you were to go through one by one and label some real numbers with natural number, you would never run out. But the interesting thing comes when you start with the assumption that you have labelled every real number with a natural number. Turns out there's always some left over...

There is a famous proof about this by Georg Cantor: http://en.wikipedia.org/wiki/Cantors_diagonal_argument

[tambourine_man]

It's only a proof if you take a leap of faith and accept certain things. You could argue that that's how math works, but to me the less axioms we need, the better.

I really love this quote from Wittgenstein:

"Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one".

http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_the...

[pavel_lishin]

Which assumptions do you find troubling?

[tambourine_man]

That although both are infinite, there are actually more of one than the other.

That's treating infinite as if it's a limit, when it's not.

If it's uncountable and unmeasurable, than it's non hierarchical.