6 ms·

A small nit: > As John Green (or Georg Cantor) taught us, some infinities are bigger than others—and the number of things is a really big infinity I don't thi

A small nit: > As John Green (or Georg Cantor) taught us, some infinities are bigger than others—and the number of things is a really big infinity I don't think this statement is true, at least not in the context it's given. At most, we'd only be able to think of countably many things, which is the smallest infinity.

sreekotay 4y ago
Is that right? So if a set is infinite but a provably strict subset of another - would we not say that set/infinity is smaller?
- elsjaako 4y ago
  Surprisingly, no. For example, even though every whole number is also a rational number, mathematicians would say the size (or more accurately, the cardinality) of the set of whole numbers is the same as that of the set of rational numbers. I'm personally a fan of the Infinite Hotel Paradox as an introduction to the subject.
- petesergeant 4y ago
  There are the same number of even integers as there are even and odd integers
  - charcircuit 4y ago
    If they are the same number what number is it.
    - tsimionescu 4y ago
      It's called ℵ 0 (Aleph 0).
      - elcomet 4y ago
        I like when someone asks a question that looks snarky, maybe rhetorical like saying "what you are saying is nonsense, look, this question cannot be answered", but then there is a perfectly valid answer
        thaumasiotes 4y ago
        The answer isn't all that valid. ℵ₀ is just a name defined by the statement "the number of integers is ℵ₀". You could also call it Bob. If you don't think Bob would be a valid name, you should reject ℵ₀ too. The name isn't unique either; by definition, ℵ₀ is equal to ℶ₀. This should be a clue that the term ℵ₀ is not actually meant to identify the number in question. Rather, what's going on is that there is a conceptual system of ℵ numbers, and another conceptual system of ℶ numbers, and the number at index 0 in each of those systems is the cardinality of the naturals.
        PebblesRox 4y ago
        But I don't think Bob is a valid name because if I use it to refer to the cardinality of the naturals, nobody would know what I'm talking about. I don't see why I have to reject aleph null just because I reject Bob.
        mbg721 4y ago
        If it were called "Bob", students would inevitably read their profs' handwriting as "eight hundred six", leading to all sorts of hilarious confusion.
        elcomet 4y ago
        > ℵ₀ is equal to ℶ₀ That's the definition of ℶ₀. So of course they are equal. "Two" is also just a name for the successor of 1. I could also call it "bob" and thus "two" would not be unique, but I don't see the point. The fact is that the cardinal of countable numbers is a mathematical concept which has a name, and can be manipulated. Which is what matters, and what the parent poster maybe did not understand.
- puffoflogic 4y ago
  Not only is the answer "no" like the sibling comment says, but in fact one definition of an infinite set is that it can be put into one-to-one correspondence with a strict subset of itself. In other words, infinite sets are precisely those for which your concept of size doesn't work. Another sibling comment used the even/odd example, but that's not necessary to dispel this particular misconception. Consider the set of non-negative integers and the set of positive integers. That is, {0,1,2,3,...} and {1,2,3,4,...}. The latter is a strict subset of the former. Maybe I have just done mathematics for too long, but to me these are intuitively, "obviously" the same size. What would it even mean for one of them to be smaller? Which one is the same size as {-1,-2,-3,...}, if either of them? Even doing folk mathematics, if the size of the first is "infinity" then the size of the second is "infinity minus one which is still infinity".
- tsimionescu 4y ago
  No, because with infinities, even if a set is a subset of another, you may still be able to find one element in the first set that corresponds to every element of the other set. For example, the natural numbers are a subset of the whole numbers, but there is a natural number that corresponds to every whole number. To see this, we can order the whole numbers like this: {0, -1, 1, -2, 2, -3, 3, ...}, and we can easily see that we can now assign one natural number to each of them (0 -> 0, -1 -> 1, 1 -> 2, -2 -> 3, ...). Since you'll never run out of naturals, you won't ever find a whole number that doesn't have a corresponding natural number. Since assigning a natural number this way is equivalent to counting the elements of the other set (in this scheme, I could say that -2 is the 3rd whole number), this type of infinity is called "countable infinity". The natural numbers, the whole numbers, and the rational numbers are all countably infinite. In contrast, the irrational numbers and the real numbers are not. In fact, even the real interval [0, 1] is not countable, so this interval is considered to have more elements than N (the set of natural numbers). Note that while there is only one countable infinity, there are many uncountable infinities - so not all uncountably infinite sets are considered as large. If you're curious about this area, the study of these concepts is done via "transfinite numbers" - particularly, the transfinite cardinal numbers (there are also transfinite ordinals).
- lupire 4y ago
  You could (using "is a subset" as a partial order), but you can't make a total order. Any way you try to compare size of sets where neither is subset of the other, while preserving your intuition of "size" will run into trouble. You make "the ordinals" sort of using your idea, but that isn't really measurement of "size"; it's more like an assignment of ranks.
alexmolas 4y ago
But you can also think of any real number, such as pi or e, which came from an uncountable set.
- mr_mitm 4y ago
  > But you can also think of any real number I don't think that's true. We can certainly think of certain numbers which we gave a name to and have defined it in some way. But there are a _lot_ of real numbers. We could think about it this way: we can only describe (and therefore think of) numbers using a finite number of symbols out of a finite alphabet. That makes it only countable.
  - kzrdude 4y ago
    Can we think of any computable number? And then some uncomputable ones if we give them a name/concept. It seems quite wild
    - karatinversion 4y ago
      What does it take to think of a number? We can certainly refer to particular uncomputable numbers - choose an enumeration of Turing machines, and let x be the real number whose binary expansion has 1 at the i-th place if the i-th Turing machine halts, and 0 otherwise.
    - deleted 4y ago
      [deleted]
    - jerf 4y ago
      There are computable numbers whose minimum expression requires more bits than the universe can hold, using any possible encoding the universe has sufficient bits to define. Infinity is much bigger than any finite number. This includes even "all possible numbers reachable in all possible finite encodings of a number within a given size", as inconceivably large as that is. No matter how large that set is, somewhere, there is some maximum number that is the largest possible number you can specify under the conditions I've given here, and that number is 0% of the way along the number line, which means that 100% of all "computable numbers" are larger than than that number. There are some uncomputable numbers that have been given names. We even have the first few digits of some of them. See for instance: https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant

sreekotay 4y ago

Is that right? So if a set is infinite but a provably strict subset of another - would we not say that set/infinity is smaller?

elsjaako 4y ago
Surprisingly, no. For example, even though every whole number is also a rational number, mathematicians would say the size (or more accurately, the cardinality) of the set of whole numbers is the same as that of the set of rational numbers. I'm personally a fan of the Infinite Hotel Paradox as an introduction to the subject.
petesergeant 4y ago
There are the same number of even integers as there are even and odd integers
- charcircuit 4y ago
  If they are the same number what number is it.
  - tsimionescu 4y ago
    It's called ℵ 0 (Aleph 0).
    - elcomet 4y ago
      I like when someone asks a question that looks snarky, maybe rhetorical like saying "what you are saying is nonsense, look, this question cannot be answered", but then there is a perfectly valid answer
      - thaumasiotes 4y ago
        The answer isn't all that valid. ℵ₀ is just a name defined by the statement "the number of integers is ℵ₀". You could also call it Bob. If you don't think Bob would be a valid name, you should reject ℵ₀ too. The name isn't unique either; by definition, ℵ₀ is equal to ℶ₀. This should be a clue that the term ℵ₀ is not actually meant to identify the number in question. Rather, what's going on is that there is a conceptual system of ℵ numbers, and another conceptual system of ℶ numbers, and the number at index 0 in each of those systems is the cardinality of the naturals.
        PebblesRox 4y ago
        But I don't think Bob is a valid name because if I use it to refer to the cardinality of the naturals, nobody would know what I'm talking about. I don't see why I have to reject aleph null just because I reject Bob.
        mbg721 4y ago
        If it were called "Bob", students would inevitably read their profs' handwriting as "eight hundred six", leading to all sorts of hilarious confusion.
        elcomet 4y ago
        > ℵ₀ is equal to ℶ₀ That's the definition of ℶ₀. So of course they are equal. "Two" is also just a name for the successor of 1. I could also call it "bob" and thus "two" would not be unique, but I don't see the point. The fact is that the cardinal of countable numbers is a mathematical concept which has a name, and can be manipulated. Which is what matters, and what the parent poster maybe did not understand.
puffoflogic 4y ago
Not only is the answer "no" like the sibling comment says, but in fact one definition of an infinite set is that it can be put into one-to-one correspondence with a strict subset of itself. In other words, infinite sets are precisely those for which your concept of size doesn't work. Another sibling comment used the even/odd example, but that's not necessary to dispel this particular misconception. Consider the set of non-negative integers and the set of positive integers. That is, {0,1,2,3,...} and {1,2,3,4,...}. The latter is a strict subset of the former. Maybe I have just done mathematics for too long, but to me these are intuitively, "obviously" the same size. What would it even mean for one of them to be smaller? Which one is the same size as {-1,-2,-3,...}, if either of them? Even doing folk mathematics, if the size of the first is "infinity" then the size of the second is "infinity minus one which is still infinity".
tsimionescu 4y ago
No, because with infinities, even if a set is a subset of another, you may still be able to find one element in the first set that corresponds to every element of the other set. For example, the natural numbers are a subset of the whole numbers, but there is a natural number that corresponds to every whole number. To see this, we can order the whole numbers like this: {0, -1, 1, -2, 2, -3, 3, ...}, and we can easily see that we can now assign one natural number to each of them (0 -> 0, -1 -> 1, 1 -> 2, -2 -> 3, ...). Since you'll never run out of naturals, you won't ever find a whole number that doesn't have a corresponding natural number. Since assigning a natural number this way is equivalent to counting the elements of the other set (in this scheme, I could say that -2 is the 3rd whole number), this type of infinity is called "countable infinity". The natural numbers, the whole numbers, and the rational numbers are all countably infinite. In contrast, the irrational numbers and the real numbers are not. In fact, even the real interval [0, 1] is not countable, so this interval is considered to have more elements than N (the set of natural numbers). Note that while there is only one countable infinity, there are many uncountable infinities - so not all uncountably infinite sets are considered as large. If you're curious about this area, the study of these concepts is done via "transfinite numbers" - particularly, the transfinite cardinal numbers (there are also transfinite ordinals).
lupire 4y ago
You could (using "is a subset" as a partial order), but you can't make a total order. Any way you try to compare size of sets where neither is subset of the other, while preserving your intuition of "size" will run into trouble. You make "the ordinals" sort of using your idea, but that isn't really measurement of "size"; it's more like an assignment of ranks.

alexmolas 4y ago

But you can also think of any real number, such as pi or e, which came from an uncountable set.

mr_mitm 4y ago
> But you can also think of any real number I don't think that's true. We can certainly think of certain numbers which we gave a name to and have defined it in some way. But there are a _lot_ of real numbers. We could think about it this way: we can only describe (and therefore think of) numbers using a finite number of symbols out of a finite alphabet. That makes it only countable.
- kzrdude 4y ago
  Can we think of any computable number? And then some uncomputable ones if we give them a name/concept. It seems quite wild
  - karatinversion 4y ago
    What does it take to think of a number? We can certainly refer to particular uncomputable numbers - choose an enumeration of Turing machines, and let x be the real number whose binary expansion has 1 at the i-th place if the i-th Turing machine halts, and 0 otherwise.
  - deleted 4y ago
    [deleted]
  - jerf 4y ago
    There are computable numbers whose minimum expression requires more bits than the universe can hold, using any possible encoding the universe has sufficient bits to define. Infinity is much bigger than any finite number. This includes even "all possible numbers reachable in all possible finite encodings of a number within a given size", as inconceivably large as that is. No matter how large that set is, somewhere, there is some maximum number that is the largest possible number you can specify under the conditions I've given here, and that number is 0% of the way along the number line, which means that 100% of all "computable numbers" are larger than than that number. There are some uncomputable numbers that have been given names. We even have the first few digits of some of them. See for instance: https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant