4 ms·

Along with convergence not being identity, any time you start playing with infinity, and then pretend that (∞-1 = ∞) you know that you are looking at a proof wi

Along with convergence not being identity, any time you start playing with infinity, and then pretend that (∞-1 = ∞) you know that you are looking at a proof with problems…

RiderOfGiraffes 16y ago
Provided it's done with care, the argument that removing a point from a countable set still leaves a countable set is perfectly valid. The trick is knowing that sometimes care is required. There are many proofs that rely on removing finitely many points from an infinite set still leaving an infinite set. The definition of division on the equivalence classes of Cauchy sequences in the one of the definitions of the real numbers is just one example.
- brown9-2 16y ago
  Sorry, I'm not understanding a part of this statement and it sounds like you know math a whole lot better than I do: infinity is a "countable set"? Could you explain this more?
  - snark 16y ago
    A "countable set" in that it has the same cardinality as the countable numbers (1, 2, 3, ...). For instance, the primes are countably infinite; you can map the set of primes onto the countable numbers (the first prime number is 2, the second is 3, the third is 5, ...). The term comes from Georg Cantor's work in the late 19th century. Notably, while the set of rational numbers is countable, the reals (including transcendental numbers such as e and pi) are not; there's a fairly simply proof using diagonalization that you can Google, although that's not how Cantor originally demonstrated it.
  - RiderOfGiraffes 16y ago
    It stems from what you mean by "infinity". Most people use the term as if infinity is a number, and that gets you into trouble. You need to say what you mean. He talked about \infty - 1, and really that means - "take an infinite-sized set and remove an element from it." Now you have the question: Which infinite-sized set? In particular, there are different sizes of infinite sized sets. The set of positive whole numbers, the set of primes, and the set of rationals are all "the same size" (under a reasonable definition of the term). However, the set of real numbers, or the set of subsets of primes, are both "bigger" (in some very real sense). So I was responding to the comment about "\infty - 1 being \infty" causing trouble. When you treat these things properly (whatever that means) then it doesn't cause trouble. At least, not any more, and not for people who understand the care required. tl;dr : If you mention "infinity" you are probably talking about the size of a set.
    - lmkg 16y ago
      This gets into something that you and I discussed earlier, that "infinity" can refer to two different concepts. While your interpretation is well-defined, and reasonable, it's not what people usually mean when they think of that expression. Even knowing the math and notation behind both of them interpretations and the difference between them, I still tend to see \inf-1 as an ill-defined arithmetic (algebraic) expression rather than a well-defined set operation. Note also, that you can treat \inf as an ordinal number, rather than a cardinal or a limit point, and subtracting one as the inverse of the ordinal successor operation. In this case, \inf-1 has a rigorous definition, which exists almost all of the time, and is somewhat closer to the naive, ill-defined algebraic interpretation. Under this definition, \inf-1 does not equal \inf.

Along with convergence not being identity, any time you start playing with infinity, and then pretend that (∞-1 = ∞) you know that you are looking at a proof with problems…

RiderOfGiraffes 16y ago

Provided it's done with care, the argument that removing a point from a countable set still leaves a countable set is perfectly valid. The trick is knowing that sometimes care is required. There are many proofs that rely on removing finitely many points from an infinite set still leaving an infinite set. The definition of division on the equivalence classes of Cauchy sequences in the one of the definitions of the real numbers is just one example.

brown9-2 16y ago
Sorry, I'm not understanding a part of this statement and it sounds like you know math a whole lot better than I do: infinity is a "countable set"? Could you explain this more?
- snark 16y ago
  A "countable set" in that it has the same cardinality as the countable numbers (1, 2, 3, ...). For instance, the primes are countably infinite; you can map the set of primes onto the countable numbers (the first prime number is 2, the second is 3, the third is 5, ...). The term comes from Georg Cantor's work in the late 19th century. Notably, while the set of rational numbers is countable, the reals (including transcendental numbers such as e and pi) are not; there's a fairly simply proof using diagonalization that you can Google, although that's not how Cantor originally demonstrated it.
- RiderOfGiraffes 16y ago
  It stems from what you mean by "infinity". Most people use the term as if infinity is a number, and that gets you into trouble. You need to say what you mean. He talked about \infty - 1, and really that means - "take an infinite-sized set and remove an element from it." Now you have the question: Which infinite-sized set? In particular, there are different sizes of infinite sized sets. The set of positive whole numbers, the set of primes, and the set of rationals are all "the same size" (under a reasonable definition of the term). However, the set of real numbers, or the set of subsets of primes, are both "bigger" (in some very real sense). So I was responding to the comment about "\infty - 1 being \infty" causing trouble. When you treat these things properly (whatever that means) then it doesn't cause trouble. At least, not any more, and not for people who understand the care required. tl;dr : If you mention "infinity" you are probably talking about the size of a set.
  - lmkg 16y ago
    This gets into something that you and I discussed earlier, that "infinity" can refer to two different concepts. While your interpretation is well-defined, and reasonable, it's not what people usually mean when they think of that expression. Even knowing the math and notation behind both of them interpretations and the difference between them, I still tend to see \inf-1 as an ill-defined arithmetic (algebraic) expression rather than a well-defined set operation. Note also, that you can treat \inf as an ordinal number, rather than a cardinal or a limit point, and subtracting one as the inverse of the ordinal successor operation. In this case, \inf-1 has a rigorous definition, which exists almost all of the time, and is somewhat closer to the naive, ill-defined algebraic interpretation. Under this definition, \inf-1 does not equal \inf.