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There are not only different sizes of infinity, as this article is trying to show, but also different types of infinity? In fact, I'm coming to believe that th

There are not only different sizes of infinity, as this article is trying to show, but also different types of infinity? In fact, I'm coming to believe that the usual methods of teaching infinity are wrong (not the concepts, just the presentation). People have this intuition that there can't be anything bigger than infinity. We reinforce that by showing that the rationals are countable, that Q^2 is countable, in fact Q^n is countable. Even the algebraics are countable. We create the intuition that anything infinite is countable. Then we present 2^N and everything goes bizarre. Some people like that, but others decide that it's all meaningless. I think this method of presentation does the majority of people a disservice. But not only is there the set-theoretic infinity, there is also the geometric infinity. That actually better matches people's intuition. Separating the ideas of the set theoretic and the geometric versions of infinity has, I'm finding, huge benefits when trying to help people understand what's going on. Of course, there's also cardinal versus ordinal infinities as well. That's fun.

tokenadult 17y ago
Separating the ideas of the set theoretic and the geometric versions of infinity has, I'm finding, huge benefits when trying to help people understand what's going on. Expand on that please, so I can make sure I understand what you are writing here.
- RiderOfGiraffes 17y ago
  It might have been you that put me onto this - I did get the idea from HN, but haven't had time to track down the original suggestion. This is a brief reply - I don't have time for more detail now, I am intending to write this up, but it's about 50th on a very long "To Do" list. People have this concept that infinity is kind of, well, "out there", as far as you can go. You can't go any further, it's all there is. People talk about parallel lines "meeting at infinity" and "there is no last point on the line" sort of thing. And that's what their intuition is telling them. There's nothing beyond infinity. Then when we talk about the cardinals we tend to reinforce that by showing that the odds, evens, squares and primes all have the same cardinality. That's a bit weird for people, but they're getting the idea that infinity is odd, but they can cope with odd. Then we show that N^2 and Q are both countable, can be put in one-one correspondence with N. They're getting cool with that too. We continue to reinforce the idea that there's infinity, and every time we do something we get the same infinity. But that's now what they expect. Infinity is kind of as far as you can go. We're matching the set-theoretic cardinals to their geometrical intuition of "out there." No wonder they get confused, upset, and occasionally angry when we then introduce 2^N as being "bigger." I'm finding that talking about geometry and the concept of infinity, then talking about sets and the concept of intinity as a different thing, despite the same name is helping people to create different models in their minds. These different models then help them not use the same intuition, and they don't get confused in the same way. Then, as the final modification, I don't start by showing that loads and loads of things are all countable. I talk about "same size" as being 1-1 matching, and discuss the idea that there could be something "bigger than N." I do that first, and then go hunting, pointing out from the beginning that historically people found this difficult. Now they get excited when I show them that 2^N is uncountable. They've been primed to want to find it, and their intuition about "infinity" isn't challenged. That's an incomplete summary - I hope it helps. Ask for more, or email me if it's not clear.
  - TallGuyShort 17y ago
    I think I agree with you. I've always just thought of infinity as having no end - and that's helped me understand all these paradoxes. For any N, 2N is certainly bigger, even if they're both so big I could never measure either. In my mind, infinity follows all the rules of mathematics that other numbers do (i.e. multiplying inifinity by two ensures that it's even, IMO), it's just that it has no end. Parallel lines don't meet at infinity - they have no meeting point.
    - RiderOfGiraffes 17y ago
      When you write 2N do you mean "2 times N"? If you do, then your intuition is at odds with mathematics as generally practised, because 2*aleph_0 is aleph_0. Further, two times infinity is the same as three times infinity, and there is no concept of infinity being even or odd. Claiming that 2 times infinity is even can lead to contradictions and inconsistencies. From that point of view, it would appear that your intuitions are not helping you. More, under some models of geometry it makes a lot of sense to say that parallel lines really do meet at infinity. It makes some theorems a lot easier to state, and easier to prove. So you may actually agree with a lot of what I'm saying, but your arguments to support that claim appear, at least on the surface, to be wrong. It may be that you have some understanding of these "paradoxes," but your other statements suggest that your inderstanding is not that of current or classical mathematics.
    - deleted 17y ago
      [deleted]
  - btilly 17y ago
    No wonder they get confused, upset, and occasionally angry when we then introduce 2^N as being "bigger." As a closet constructivist I feel compelled to point out that 2^N certainly has a more complicated internal structure, but "bigger" is entirely a question of interpretation. There are a countable number of possible constructions of something in 2^N, and therefore in the traditional interpretation of mathematics all but a countable number of members of 2^N are unconstructable. However a constructivist looks at this and asks what sense it makes to talk about the real "existence" of mathematical objects that cannot be represented in any way, shape or form, even in principle? To a constructivist these don't really exist at all, so there really can't be more of them. Incidentally the classic picture most mathematicians have of the infinite cardinals being nicely sorted by size depends on the axiom of choice. Specifically Friedrich Hartogs proved in 1915 that if all infinite sets are comparable in size, then the axiom of choice holds. The reverse statement is also true, and is easy to prove from the well-ordering principle. So if you want to be a classical mathematician but are heretic enough to doubt the axiom of choice, then again the simple picture people have of infinite cardinals breaks down.

tokenadult 17y ago

Separating the ideas of the set theoretic and the geometric versions of infinity has, I'm finding, huge benefits when trying to help people understand what's going on. Expand on that please, so I can make sure I understand what you are writing here.

RiderOfGiraffes 17y ago
It might have been you that put me onto this - I did get the idea from HN, but haven't had time to track down the original suggestion. This is a brief reply - I don't have time for more detail now, I am intending to write this up, but it's about 50th on a very long "To Do" list. People have this concept that infinity is kind of, well, "out there", as far as you can go. You can't go any further, it's all there is. People talk about parallel lines "meeting at infinity" and "there is no last point on the line" sort of thing. And that's what their intuition is telling them. There's nothing beyond infinity. Then when we talk about the cardinals we tend to reinforce that by showing that the odds, evens, squares and primes all have the same cardinality. That's a bit weird for people, but they're getting the idea that infinity is odd, but they can cope with odd. Then we show that N^2 and Q are both countable, can be put in one-one correspondence with N. They're getting cool with that too. We continue to reinforce the idea that there's infinity, and every time we do something we get the same infinity. But that's now what they expect. Infinity is kind of as far as you can go. We're matching the set-theoretic cardinals to their geometrical intuition of "out there." No wonder they get confused, upset, and occasionally angry when we then introduce 2^N as being "bigger." I'm finding that talking about geometry and the concept of infinity, then talking about sets and the concept of intinity as a different thing, despite the same name is helping people to create different models in their minds. These different models then help them not use the same intuition, and they don't get confused in the same way. Then, as the final modification, I don't start by showing that loads and loads of things are all countable. I talk about "same size" as being 1-1 matching, and discuss the idea that there could be something "bigger than N." I do that first, and then go hunting, pointing out from the beginning that historically people found this difficult. Now they get excited when I show them that 2^N is uncountable. They've been primed to want to find it, and their intuition about "infinity" isn't challenged. That's an incomplete summary - I hope it helps. Ask for more, or email me if it's not clear.
- TallGuyShort 17y ago
  I think I agree with you. I've always just thought of infinity as having no end - and that's helped me understand all these paradoxes. For any N, 2N is certainly bigger, even if they're both so big I could never measure either. In my mind, infinity follows all the rules of mathematics that other numbers do (i.e. multiplying inifinity by two ensures that it's even, IMO), it's just that it has no end. Parallel lines don't meet at infinity - they have no meeting point.
  - RiderOfGiraffes 17y ago
    When you write 2N do you mean "2 times N"? If you do, then your intuition is at odds with mathematics as generally practised, because 2*aleph_0 is aleph_0. Further, two times infinity is the same as three times infinity, and there is no concept of infinity being even or odd. Claiming that 2 times infinity is even can lead to contradictions and inconsistencies. From that point of view, it would appear that your intuitions are not helping you. More, under some models of geometry it makes a lot of sense to say that parallel lines really do meet at infinity. It makes some theorems a lot easier to state, and easier to prove. So you may actually agree with a lot of what I'm saying, but your arguments to support that claim appear, at least on the surface, to be wrong. It may be that you have some understanding of these "paradoxes," but your other statements suggest that your inderstanding is not that of current or classical mathematics.
  - deleted 17y ago
    [deleted]
- btilly 17y ago
  No wonder they get confused, upset, and occasionally angry when we then introduce 2^N as being "bigger." As a closet constructivist I feel compelled to point out that 2^N certainly has a more complicated internal structure, but "bigger" is entirely a question of interpretation. There are a countable number of possible constructions of something in 2^N, and therefore in the traditional interpretation of mathematics all but a countable number of members of 2^N are unconstructable. However a constructivist looks at this and asks what sense it makes to talk about the real "existence" of mathematical objects that cannot be represented in any way, shape or form, even in principle? To a constructivist these don't really exist at all, so there really can't be more of them. Incidentally the classic picture most mathematicians have of the infinite cardinals being nicely sorted by size depends on the axiom of choice. Specifically Friedrich Hartogs proved in 1915 that if all infinite sets are comparable in size, then the axiom of choice holds. The reverse statement is also true, and is easy to prove from the well-ordering principle. So if you want to be a classical mathematician but are heretic enough to doubt the axiom of choice, then again the simple picture people have of infinite cardinals breaks down.