7 ms·

Which domain? Reals? That's a lot of numbers. Infinity, in fact.

• pavel_lishin 15y ago
  Not just an infinity, but a bigger infinity than the integers.
  • tambourine_man 15y ago
    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 15y ago
      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 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 15y ago
        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 15y ago
          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 http://en.wikipedia.org/wiki/Cantors_diagonal_argument
          • tambourine_man 15y ago
            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_theory http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_the...
            • pavel_lishin 15y ago
              Which assumptions do you find troubling?
              • tambourine_man 15y ago
                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.
                • pavel_lishin 15y ago
                  > That although both are infinite, there are actually more of one than the other. That's not the assumption, that's the result.
• tokenadult 15y ago
  Heck, there are infinite prime numbers, all of which are natural numbers. http://primes.utm.edu/notes/proofs/infinite/euclids.html http://primes.utm.edu/notes/proofs/infinite/euclids.html http://mathworld.wolfram.com/EuclidsTheorems.html http://mathworld.wolfram.com/EuclidsTheorems.html http://www-users.cs.york.ac.uk/susan/cyc/p/primeprf.htm http://www-users.cs.york.ac.uk/susan/cyc/p/primeprf.htm http://primes.utm.edu/notes/proofs/infinite/ http://primes.utm.edu/notes/proofs/infinite/