Are there too many numbers?

Answer 1

Way too many. I mean, sure it was fine with small numbers like 1, 2, 10, 100, but then we started getting into large numbers like the thousands, and the millions, and even the billions. Then someone figured out the idea of exponents and we got numbers like the googol (not to be confused with google), which exceed even the number of atoms in the entire universe. But even this wasn't enough for the mathematicians, they had to have more -- more than any preassigned quantity. And so now they're telling us that we have infinitely many numbers! Infinitely many! And they didn't stop there -- no, then they created the rational numbers and we got infinity times infinity -- although the Cantorians reassure us that's not actually any bigger than the first infinity.

And yet, even after creating the rationals they still wouldn't stop!! "There are gaps in the number line" they said. "It must be completed" they said. "Any nonempty set bounded above _must_ have a supremum," they said. And so they created an infinity that WAS bigger than the natural numbers, and they called this impossible construct -- get this -- the "real" numbers. As if there's anything real about numbers!

And they still wouldn't stop. Remember those Cantorians I told you about earlier? They went on and created this idea that there are infinitely many infinities, and created a whole system of cardinal numbers to organize them. Do you have any idea how many cardinals there are? It's not like you have only, say, countably many infinities, or even as many infinities as there are real numbers. No, you take any one of the possible degrees of infinities -- any of the transfinite alephs -- and there are more cardinal numbers than that. There are so many, they can't even create a set to hold them all. That's right -- the cardinal numbers form a proper class! When Cantor himself first discovered them (well, okay, he was working with the ordinals, but the two classes are isomorphic), he called them an "absolutely infinite _inconsistent_ multiplicity". Inconsistent!!! The cardinal (pardon the pun) sin in mathematics is inconsistency, but instead of shying away from them like sane people, they embraced the contradiction, saying that it was a proof that the cardinals could not be gathered into a set, and restricting our notion of set only to those things that could be constructed from an inductive set using certain functions (union, powerset, and replacement, plus the ability to select an arbitrary choice set). And this patchwork they told us was consistent, and was fit to be a foundation for all mathematics, despite the fact that they knew they could not prove that on pain of Godel's second incompleteness theorem (well, unless ZFC is actually inconsistent, in which case it trivially proves its own consistency and the theorem is merely false).

And you want to hear something really scary?

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They _still_ haven't stopped. You don't hear it often, but in the dimly lit halls of universities you will hear the set theorists whispering about so-called "large cardinals," which are so massive that their existence is not even provably _consistent_ with set theory, even under the assumption that set theory itself is consistent. They continue to study these imagined constructs, far beyond human comprehension, in the hope of discovering the ultimate secrets of mathematics. If you ask me, they're crazier than the people who tried to make the Philosopher's stone.



P.S. -- If any set theorists are reading this, the above account was meant to be tongue in cheek. I don't actually think all set theorists are crazy.

Answer 2

There are only 10 basic numbers 0,1,2,3,4,5,6,7,8,9. Every other number derives from those ten.

Answer 3

There is a sufficient number of numbers. We couldn't do with one less or one more.

Answer 4

Nope!

10 fingers

10 toes

That's all I need!

Answer 5

way too many.

Answer 6

I reckon not!