To reply to what a few have said; there are several different meanings for infinity. The one I have been talking about so far is the most common colloquial meaning, which is the size of the set of counting numbers, {1, 2, 3, ... }. Another meaning of infinity is that found in analysis, where it is used as shorthand for statements like, '1/x tends to infinity as x tends to 0 from above', but this isn't actually about any entity 'infinity', as x never actually reaches 0, it's just a statement about the way that 1/x is behaving for non-zero values of x. The last use that I can think of for infinity in maths is a formal use, where for instance you might want to say that actual division of 1 by 0 is okay, so you just add some new point to the set, which you call 'infinity' for colloquial reasons, and then add some rules for how it behaves. This last usage is just formalism, you could just as well label the new point 'Garry' rather than 'infinity'; wouldn't make any difference.
The point is that for all three of these different meanings, they each have very precise definitions, and the definitions are about simple and non-contradictory things.
 Originally Posted by tempusername
Are you telling me that anything, even the concept of zero, even "nothing," mathematically falls into a set?!
You can take different approaches, and it's kind of too philosophical and/or meaningless for most mathematicians to worry about, but if you're aiming to put mathematics on a highly formal, minimal basis, that basis tends to be sets, yes. I'm not sure there is anything in mathematics that couldn't in principle be reduced to some statement of set theory. Some of the ways it is done are quite cool... for instance, in modern mathematics, the real, continuous number line is formulated in terms of sets (using things called Dedekind cuts, which are basically just a splitting into two sets of the fractions, and fractions in turn are basically just a set of pairs of natural numbers, with some rules).
Size... d-do sets even truly have the property of size? Any property of a set is defined by the elements which it holds... So:
Counting is about the sizes of sets.
Sizing sets is about maps.
Mapping is about distinguishing elements.
Numbers distinguish.
And the only thing to do with numbers is count them... f-f-full circle...
It doesn't really come full circle. It goes sets -> mappings -> number. You distinguish mappings without reference to number.
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