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How big is infinity?  How should we think about infinity?  Can we think about infinity in sensible ways that work with the rest of mathematics?  Are all infinities the same? When a math result requires dealing with infinity, is it necessarily nonsensical?

People have been thinking about these sorts of questions since at least the time of ancient Greece.  But their thinking was confused and contradictory until Georg Cantor came on the scene around 100 years ago.  Some of what he said is strange, and he himself was certainly strange, but his methods for dealing with infinity solve paradoxes that had confused people for at least 2,000 years.

More below the fold

How many positive integers are there?  There are an infinite number.  You can tell this pretty simply, because there is no `biggest' number.  However big a number you name, I can just add one, or 10, or a million to it and get a bigger number.

How many even positive numbers are there? (The even numbers are those that are evenly divisible by 2 --- 2, 4, 6, 8, and so on). Well, there's no biggest even number, either, so there must be an infinite number of them, as well.  Similarly, there must be an infinite number of odd numbers.  But wait! That means that infinity plus infinity is infinity!  WTF?

How about rational numbers?  Surely there must be more rational numbers than integers!  After all, between any two integers there are an infinite number of rational numbers.  For instance, between 1 and 2 we have 1 1/2.  But between 1 and 1 1/2 we have 1 1/4, and so on.  But what could be bigger than infinity?  WTF?

Cantor straightened all this out.  The way he did it was to go back
to the most elementary mathematical idea --- one-to-one
correspondence.  Two sets of objects, he said, have the same size if
you can put the two sets into one-to-one correspondence.  What's
that? You may not have heard of it by those terms, but you've known
about it since you were a little kid.  Suppose you have a bunch of
apples, and a bunch of people.  Are there more apples, or people?
Suppose there's a LOT of each.  You don't want to count them all.
So, just have each person take one apple.  If there are apples left
over, there were more apples than people.  If there are people left
over, there were more people than apples.  If nothing is left over,
then the two sets were equal.

Cantor applied this idea to infinite sets.  If two sets can be put
into one-to-one correspondence, they are equal.  If not, not.

Let's apply this to the problems above:

Positive integers:  1, 2, 3, 4....
Even integers:      2, 4, 6, 8....

It looks like they match up, but we can show that they always will:
For any positive integer x, the matching even integer is 2x. Similarly, for any positive integer x, we can find a matching odd integer 2x+1.  So, weird as it seems, there are the set of even numbers is the same size as the set of integers.

We're just getting started!

Cantor also showed that the set of rational numbers is the same size.  This was a little trickier.  The rational numbers, you will remember (possibly from my diary Numbers are those that can be expressed as ratios, or fractions.  Cantor came up with a way of numbering all of these.  First, think of adding the numerator and denominator.  Then, for each total, list them in terms of size of the numerator.  Like this

0/0
0/1
0/2 1/1
0/3 1/2  2/1
0/4 1/3 2/2 3/1
and so on

eventually, you will get to every rational number.

So, are all infinite sets of numbers the same size?  Nope.  Here's where Cantor really got clever.   He did the following:

First, he noticed that any real number can be shown as a decimal. Some will be repeating decimals (like 1/3 = 0.3333....), some will be terminating decimals (like 1/2 = 0.50000....) and some will be neither (pi = 3.1415....
Next, he thought that, if there were a countable number of these, he could list them all.  Something like this

 0.500....
 0.125....
 0.314....

and so on (forever)

Then, he thought, what would happen if you read down the diagonal, reading the first digit first number, the second of the second, and so on, and then adding one to each digit? For the above, you'd get 0.635....  Now, this number can't be anywhere on the list. It can't be the first number, because the first digit doesn't match. It can't be the second, because the second digit doesn't match, and so
on.  No matter how many numbers you list, you always leave some out.

So, there are more irrational numbers than rational numbers.

Weird.  Because, no matter how close two rational numbers are, you can always find another rational number in between. But you can find even more irrational numbers.

This diary was just an intro to this stuff.  If people are interested, I can write more.  A good (if quirky) book on this material is David Foster Wallace's Everything and More (but see the comment below by danielbiss)

Originally posted to plf515 on Fri Aug 04, 2006 at 07:50 AM PDT.

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Comment Preferences

  •  tip jar (19+ / 0-)

    OK, this diary did not have anything to do with Lamont or Lieberman, but at least it wasn't meta or snark!

    (and no mention of I L or P, either)

    tips, comments, thoughts, both for this diary and the series are welcome.

    Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

    by plf515 on Fri Aug 04, 2006 at 07:42:54 AM PDT

    •  love the diary (5+ / 0-)

      and the series.  A pet peeve, though: I think Everything & More is a dangerous book to recommend to a beginner; Wallace gets an awful lot of stuff badly wrong, and if you don't know the material in advance, it's hard to figure out what he means (and in fact most readers' natural conclusion is that they themselves aren't getting it, rather than that he's mistaken).  As a mathematician (whose interest in math was awakened by this very material) and giant DFW fan, I was very disappointed to find myself largely in agreement with Michael Harris's review in the Notices of the American Mathematical Society.

      •  Thanks for the warning (4+ / 0-)

        I'm just an amateur myself, and I admit that was my reaction to some of the book.  Do you have other recommendations?

        I will update the diary to reflect this

        Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

        by plf515 on Fri Aug 04, 2006 at 07:59:59 AM PDT

        [ Parent ]

        •  Wow (1+ / 0-)
          Recommended by:
          lgmcp

          Very impressive for an amateur.

          No, unfortunately I don't have other recommendations, at least not in that exact genre.  I was really eager to read E&M precisely because I'd always felt that such a book should exist but I didn't know of one...

          •  Thanks (2+ / 0-)
            Recommended by:
            ignorant bystander, lgmcp

            I am a statistician by profession, but a practical data analyst than a theoretician.  I had a couple semesters of Calculus (Calc 1 with a guy who could teach, but couldn't speak English, Calc 2 with a guy who could speak English, but couldn't teach) and have read a lot on my own.

            If you (or anyone) wants to write something in this series, just let me know.

            Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

            by plf515 on Fri Aug 04, 2006 at 08:38:33 AM PDT

            [ Parent ]

        •  In search of infinity (3+ / 0-)
          Recommended by:
          danielbiss, lgmcp, plf515

          One I really like is "In search of infinity" by N. Ya. Vilenkin (translated by Abe Shenitzer).  See the description at Amazon, for example.   The book claims to be

          ...a popular account of the roads followed by human thought in attempts to understand the idea of the infinite in physics and in mathematics.  It tells the reader about fundamental concepts of set theory, about its evolution, and about the relevant contributions of Russian scientists.  The book is meant for a broad range of readers who want to know how the notion of the infinite has changed in time, what one studies in set theory and what the present state of the theory is.

    •  Very nice (1+ / 0-)
      Recommended by:
      plf515

      Too late for me to recommend it, but I just wanted to say so.

      Fake Canadians are total hosers.

      by theran on Mon Aug 07, 2006 at 12:01:39 AM PDT

      [ Parent ]

  •  Interesting, but is it math? (3+ / 0-)
    Recommended by:
    lgmcp, plf515, 73rd virgin

    I know even Einstien encouraged Cantor (if I am not confusing my mathematicians--maybe it was Godel), but I sorta feel like, when it comes to infinity, it's the mathematical equivalent of "how many angels can dance on the head of a pin."

    That is, it is interesting, but is it of an real consequence? It seems to me, it can't be tested; nor can it be put to any practical use.

    Of course, once upon a time prime numbers had no practical use, and then along came cryptology. But I think infinity is like dragons and fairies--not real.

    Or am I wrong?

    •  Einstein was friends with Godel (5+ / 0-)

      see my earlier diary in this series.Proof, certainty and Godel

      As for this being math.....well, if it isn't math, what is it?  The test of whether something is math or not is certainly not whether it is of practical use (indeed, Hardy would think that anything of practical use was NOT really math).  

      Is infinity real?  Well....are numbers real?  You've never seen or heard or felt 2.  Two OF something, sure.  but not 2 itself.

      Bertrand Russell once said

      math is the only subject in which we never know what we are talking about, nor whether what we are saying is true.

      Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

      by plf515 on Fri Aug 04, 2006 at 08:06:40 AM PDT

      [ Parent ]

      •  Hmm, I'd say... (2+ / 0-)
        Recommended by:
        lgmcp, plf515

        ...that the neat thing about math is that you can know that what you're saying is true, even though you don't know what you're talking about.

        As for applicability, isn't the distinction between countable and uncountable infinities essential to the mathematics used in modern physics?

    •  Conflating questions a little bit (2+ / 0-)
      Recommended by:
      lgmcp, plf515

      Whether or not it's math is a very different question than what's its practical use; it's math because, well, roughly because it's about numbers/patterns/axioms, and it's tested in that it fits seamlessly into the internally consistent body of mathematical ideas.

      It's a little hard for me to give a good honest answer to the "practical use" question, because these ideas have become the underpinning of so much other mathematics (including lots of aspects of the study of prime numbers, to use your example) that it's hard to imagine much of today's practical mathematics happening without them.  But I don't have at my fingertips a direct practical application.  I'll think about it some more though...

      (Btw I'd guess you mean Godel, who was a little younger than Einstein and a colleague of his in Princeton, not Cantor, who was enough older that I doubt they met.)

      •  Infinity has at least one practical use (1+ / 0-)
        Recommended by:
        ek hornbeck

        doesn't it?  In that it's where asympotes approach too when they're not approaching zero?

        Seems pretty foundational to calculus and so on, as far as I can tell.  

        The extinction of the human race will come from its inability to EMOTIONALLY comprehend the exponential function -- Edward Teller.

        by lgmcp on Fri Aug 04, 2006 at 08:50:29 AM PDT

        [ Parent ]

        •  asymptotes yes, but... (1+ / 0-)
          Recommended by:
          plf515

          The very idea of an asymptote involves infinity, whether the asymptote exists (i.e. is some particular finite number) or doesn't.  So yes, if you think asymptotes are useful (and I do, calculus as you point out is based on them and calculus is very useful when you are building bridges) then infinity is a useful idea.

          There is more than one way for an asymptote to fail to exist, however, and only some of them have to do with approaching infinity.  f(n) = n could be said to have asymptote infinity as n increases (although there are always exactly the same number of integers between n and infinity no matter what n is), however f(n) = sine(n) is always between -1 and 1, and doesn't approach any particular number more frequently than any other in that range as n increases.  So sine(n) has no asymptote and never gets anywhere near infinity.

    •  absolutely (2+ / 0-)
      Recommended by:
      lgmcp, plf515

      Yes, it is mathematics; in fact, by research standards, it is very basic mathematics.

      Example:  in my last research paper, my partner and I proved that there were an uncountable number of counterexamples to a perviously open conjecture, and that each of these counterexamples had an uncountable number of "singularities".  

      When liberals saw 9-11, we wondered how we could make the country safe. When conservatives saw 9-11, they saw an investment opportunity.

      by onanyes on Fri Aug 04, 2006 at 08:11:51 AM PDT

      [ Parent ]

    •  Come to think of it... (1+ / 0-)
      Recommended by:
      lgmcp

      ...I just recalled a practical reference to infinity; when things enter a queue faster than they leave, the average wait time is infinity; something every computer science graduate knows.

      And calculus depends on infinity too.

      So "never mind" my first question.

      But let me add a knew one: Did Cantor's work help with calculus, queueing theory, or anything else that uses infinity/infinitesimals?

      (Not quibbling--just curious.)

  •  Infinitely better than (8+ / 0-)

    Most diaries one reads these days.

    Thank god you didn't call it "calculate my fractions, kos!"

  •  Never liked the diagnolization proof (1+ / 0-)
    Recommended by:
    plf515

    Let's work in binary here, and to make it real easy, on the interval of [0,1).

    You list out "all" the numbers, and for each row, change the column, right?  Well, each column is another digit, and we can express 2^digit numbers in binary.  Meanwhile we remove digit numbers via changing the diagnol value.

    Incoming numbers: 2^d
    Outgoing numbers: d
    Total numbers that are in the list but not addressed: 2^d - d

    As d goes to infinity, the amount remaining in the list just explodes; an exponential like that simply overwhelms the linear subtraction.  You don't get to the end of the list because it's simply not a square matrix; it's got 2^d rows and only d columns (i.e. there is no main diagnol you can do such a method on).

    •  I am not sure I follow you (1+ / 0-)
      Recommended by:
      sprawlrat

      but I think that what you say is exactly the point:
      The list 'explodes' - it's an uncountable infinity rather than a countable one (like the integers or rationals).

      Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

      by plf515 on Fri Aug 04, 2006 at 08:42:07 AM PDT

      [ Parent ]

      •  'explodes' = shoots off to infinity at rapid rate (0+ / 0-)
        Point being you can pull the same stunt with a countable set.  I can start listing positive integers, for each digit I go out to (assuming here binary again) I have listed 2^d numbers.   Let's go out to a d of 2 and then 3 to see what happens.

        00
        01
        10
        11

        are all the binary numbers I can express with 2 digits.  Using Cantor's method, the number I come up with is 1111, a four digit number.  It's obviously in my list of positive integers, just look at it, but I just haven't listed that far yet.

        000
        001
        010
        011
        100
        101
        110
        111

        are the 3 digits.  Cantor's method yields 11,111,111: an eight digit number.  Still definately a positive integer, and is in my list, I just haven't gotten there yet.  It's the same problem we'd see going on the other side of the decimal point.  At any point in listing, the number I come up with that's "not in the list" simply requires more digits than I've used so far.

        Now, you'd call me crazy if I claimed the positive integers are uncountable.  Since we can equate them with the natural numbers (aka counting numbers), I'd have to agree with you.  What I've proved there is just the truism that 2^d > d.  More generally with base b number system: b^d > d.  You effectively remove one number for each digit (by making the number at that place different), but you put in b^d per digit.

        •  doesn't work (0+ / 0-)

          Cantor's argument only works when each sequence has an infinite number of elements

          Here is how it goes with just 0's and 1's:

          take any countable collection of sequences of zeros and ones.

          Lable them f1, f2, f3, ...and so on.
          Let fi(k) denote the "k'th" element of the "i'th" sequence.  This set of sequences is now fixed.

          Now construct a new sequence "q" where q(k) differs from fk(k) at the "k'th" element.

          Then q does not belong to the set of sequences f1, f2, ....fk...

          When liberals saw 9-11, we wondered how we could make the country safe. When conservatives saw 9-11, they saw an investment opportunity.

          by onanyes on Fri Aug 04, 2006 at 10:12:04 AM PDT

          [ Parent ]

        •  Infinite expansion, mapping to different sets (0+ / 0-)

          If you continue with the binary expansion method described above, you'll always get another integer. All that shows is that for every integer you can concoct, there exists one larger, or in other words your function's domain and range lie within the same set. Mixing together all the various kinds numbers (i.e. integers, rationals, irrationals) like Cantor does produces a slightly, but crucially different, result.

          His input set includes integers, rationals, and irrationals. The resulting value after diagonalization, however, will always be irrational if your decimal expansions go to infinity. Thus no matter what set you use to try to "count" the irrational numbers, you'll always come up short because there's at least one more that you cannot count because that number doesn't exist in your counting set, even if you include "all" of the irrational numbers.

          In other words, the "diagonalization" function when applied to all real numbers always produces an irrational number that lies outside your set. If you add the resulting number back to the set and repeat the process, you'll get another uncountable number that lies outside your set, and so on and so forth.  You will never, however, get another rational or integer value because all of your outputs are, by definition, irrational (see this Wikipedia entry for the definition of irrational numbers -- http://en.wikipedia.org/... Thus the irrational numbers comprise a set of cardinality greater than the set of real numbers.

    •  I'm also not so sure I follow (1+ / 0-)
      Recommended by:
      sprawlrat

      The fact that this 2^d by d matrix isn't square is exactly the point, but one needs to establish this for an arbitrary (i.e. possibly infinite) value of d.  I don't know how you get around using some kind of diagonalization to do that.  (Sure, you're correct that the list "explodes" but why wouldn't it explode in the same way if instead of 2^d you said d^8 or something?  That's where some kind of diagonalization is needed.)

      •  at any value of d, that will not be square (0+ / 0-)
        Unless 2^(infinity) equals (infinity).   Start picking large numbers, plug them in, and see that the exponential will always dwarf the linear term.

        The reason I picked 2^d was for number of numbers expressed at a certain number of digits.   You can replace 2 with any integer > 1 (for the number system of your choice), but 2 gives the closest to possibly working.

        •  That's the point (0+ / 0-)

          You need to prove that 2^(infinity) isn't (infinity).  Sounds obvious until you notice, as is pointed out in the diary, that, roughly speaking, (infinity)^2 = (infinity), or indeed, (infinity)^k = (infinity) for any integer k.

          Noticing that the the exponential dwarfs the linear for large numbers is a good start but I think you'll need some kind of Russell's paradox or other form of diagonalization to actually make it a complete proof in the infinite case.

    •  do you know a reference for this 'proof'? n/t (1+ / 0-)
      Recommended by:
      sprawlrat
      •  there isn't one to my knowledge (0+ / 0-)
        It really has been a "I just don't like this proof" thing since the 10th grade.  Unfortunately, any time (including collegiate level) I have brought up the objection it becomes a "Let me get back to you on that..." to which I never actually get a reply.

        Honestly, hoping a Kossack can put any concerns I have to rest.  That would actually be really awesome, because truthfully, I would bet on Cantor over myself (it seems a safe bet).

        •  What's the objection to Cantor? (0+ / 0-)

          It's just a simple reductio, as far as I can see. If the set of all real numbers in [0, 1] were countable, then they could all be put in a sequence; but given any such sequence, Cantor shows how to construct another real number in [0, 1] that isn't in the sequence. So the set of all real numbers in [0, 1] isn't countable after all.

          •  not really (1+ / 0-)
            Recommended by:
            indybend

            It's just a simple reductio, as far as I can see. If the set of all real numbers in [0, 1] were countable, then they could all be put in a sequence; but given any such sequence, Cantor shows how to construct another real number in [0, 1] that isn't in the sequence. So the set of all real numbers in [0, 1] isn't countable after all.

            Actually, all real numbers in [0,1] can be realized by sequences.

            What Cantor's arugment actually shows that if you have any countable collection of sequences, you can always find another sequence that doesn't belong to that collection.

            Hence the set of all possible sequences is uncountable
            (actually, the set of all possible sequences that have only 0 or 1 as elements is also uncountable...)

            When liberals saw 9-11, we wondered how we could make the country safe. When conservatives saw 9-11, they saw an investment opportunity.

            by onanyes on Fri Aug 04, 2006 at 10:05:03 AM PDT

            [ Parent ]

  •  As long as the room is full of math geeks.. (5+ / 0-)

    Yesterday I looked up an old friend online to see what he's been in up to in the decades since we lost touch.  The friend is Cliff Stoll .

    Turns out what he's up to now is making Klein Bottles, the glass-blowing equivalent of a Moebius strip.  Good weirdness.  

    The extinction of the human race will come from its inability to EMOTIONALLY comprehend the exponential function -- Edward Teller.

    by lgmcp on Fri Aug 04, 2006 at 08:54:16 AM PDT

  •  Countable and Uncountable (1+ / 0-)
    Recommended by:
    plf515

    There are several axiomatic ways to define numbers, and I believe Cantor was responsible for the standard set-theory way.  First integers are defined in terms of sets, then rational numbers (fractions) are defined as pairs of integers, then real numbers are defined with an idea called
    "Dedekind cuts".

    The infinity that this diary has focused on, what is called "countable" infinity, is just the beginning.  The number of rational numbers is also countably infinite, which just means there is a way to number each and every one with a positive integer.

    But the number of "real" numbers, (the ones that make up the entire number line) is larger:  there is no way to number all the real numbers using only the positive integers.  Mathematicians call the number of real numbers "uncountably" infinite.
    And there are an infinite number of other infinities, all provably different from each other, after your mind gets used to uncountable infinity.

    •  Nancy McGough on the Continuum Hypothesis (1+ / 0-)
      Recommended by:
      plf515

      ... is also a valuable source of info here, for those who want to delve further into the topic. (link)

      •  Cool (0+ / 0-)

        I didn't get into the CH in this diary, although maybe I will in another.

        I like doing the diaries, but they are a bit of a pain, because formulas have to be stored elsewhere.  
        So, each contains less than I'd like

        Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

        by plf515 on Sun Aug 06, 2006 at 04:16:50 AM PDT

        [ Parent ]

  •  Uncountably many pages (1+ / 0-)
    Recommended by:
    plf515

    Jorge Luis Borges wrote a very interesting short story based on the idea of a book with an uncountably infinite number of pages, called "The Book of Sand".  It is easy to read whether you have a background in mathematics or not, and I highly recommend it.

  •  Numbers and Games (1+ / 0-)
    Recommended by:
    plf515

    For people with a serious mathematical background who are interested in other ways to define numbers, I highly recommend Joseph Conway's book, "On Numbers and Games".  You may have heard of Conway; he came up with the computer "game" called Life.

    Conway defines all numbers, and I mean all of them, in terms of games, specifically two-player games.  Some games don't correspond to numbers, but every number, even all the infinities and ones you may not have heard of yet (the "infinitessimals" that come up in one version of the theory called the calculus of variations) all correspond to the "value" of some game.

    Conway's idea of games isn't too different than ours; chess and backgammon and the card game "War" are all good examples.  One difference is that each time a player makes a move, in Conway's theory you have a different game, where we tend to think of a game being the same game whether we are just starting or near the end.

    The value of a game can be thought of as how many moves closer to winning player A is than player B is (player A is the one who has the next move).  A game whose value is 0 is tied.

    Conway's book is split into two parts, one about games using conventional mathematics and one that proves things about numbers defined as games.  For me, there was a lot of interesting stuff about both numbers and games before the math got too hard for me to follow (I have bachelor's degrees in mathematics and computer science, and I spent a year in a graduate math program).  I definitely wouldn't recommend this book for beginners; you should be comfortable defining numbers in terms of sets, for example, before reading this book.

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