June 01, 2006

Aleph

Think about the following.

Problem 1
We have an aleph (which, I have been explained, is an infinite set) of real numbers (e.g. 1 – 2 – 3 – to infinity).

Then we have an aleph of numbers with one decimal place (e.g. 1.1 – 1.2 – 1.3 – 1.4 – to infinity).

Can we say aleph 2 is bigger than the former? To prove that, we’d have to take a chunk off, but then it wouldn’t be an aleph anymore. Can we say, then, at least, that aleph 2 is richer than aleph 1? And does that make a difference in infinity?

Problem 2
If infinity is aleph to the 1, is total nothingness then consequently aleph to the -1?

Proclaimer: This is not a revision rant, just a procrastination rant. I am strictly a student of History and of Sociology.


- 8 comments by 1 or more people

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  1. Want to know the actual answer?

    Aleph isn't an infinite set, it's the size (cardinality) of an infinite set. Aleph–0 is the size of the infinite set 0,1,2,.... The set 1.1,1.2,1.3,... has the same size aleph–0, as does 1.01,1.02,.... The set of all real numbers (with infinitely long decimal expansions) has size 2 to the power of aleph–0, which may or may not be aleph–1 (there's a long story there too).

    It's only true for infinite sets that they can be the same size as one of their subsets (so 1, 1.1, 1.2,... has 1,2,3,.... as a subset, but as infinite sets they have the same size).

    01 Jun 2006, 13:43

  2. Where does the 2 in "2 to the power of aleph –0" come from?

    And so what you're saying is that, as a whole, all alephs have the same size? But in human experience (incapable of seeing the whole of infinity) aleph 2 must be experienced as being larger.

    01 Jun 2006, 14:10

  3. you can set up a 1–1 correspondence between the 1 decimal place numbers by simply multiplying them by 10, and you can get back to the 1 decimal place numbers by dividing by 10. Since for every natural number (the numbers 1,2,3,...) you can uniquely define one of these 1 decimal place numbers, the only conclusion that makes sense is that the sets have the same size, denoted as aleph.

    The "two the power aleph" is merely convention which illustrates that one is actually "larger" in some sense than the other. Given a finite set X of numbers – eg {1, 2} – it's power set P(X) is the set of all possible subsets: { {}, {1}, {2}, {1, 2} }, where {} is the empty set. As you can see there are 2 elements in the original set, whilst there are 22 elements in the power set. It can be shown (by induction) that if a (finite) set has n elements, then its power set has 2n elements.

    Incidently, you might think the empy set is a bit of an artificial invention, but it actually provides the means to "deduce" the natural numbers to the ridiculous levels of rigour required by set theorists!

    The real numbers – any counting number, fractions etc, along with irrational numbers like square roots, and everyone's favourite ratio (pi) – can be shown to be in 1–1 correspondence with the power set of the natural numbers. So if you like, the size of the real numbers is 2^(aleph). This doesn't really have any meaning because whatever aleph is, it certainly isn't a number in the way that 4 or pi or 1/2 are numbers, but as a convention it gives the implication that one is larger than the other. Incidently, it should be mentioned that there is no such 1–1 correspondance between any set X and its power set P(X).

    By considering further power sets, you therefore end up with lots (infinitely? haha) different "infinities" – which is all quite odd and abstract. To make things slightly clearer, anything that can be paired up with the natural numbers is said to be "countably infinite," whilst each "larger" infinity is said to be "uncountably infinite." The vast majority of things people think are infinite are (perhaps disappointingly) countably infinite.

    With regards to the second problem as I mentioned before, it doesn't make any sense to perform the standard operations with aleph as you would a real number. However I suppose as a vague analogy, if you take "total nothingness" as 0, and you take any natural number n, then clearly n^–1 will become arbitrarily close to 0.

    01 Jun 2006, 20:27

  4. I have no trouble admitting that most of this is grossly beyond my grasp. But outside of mathematical theory; I debated this topic again today, and we all agreed that since humans are incapable of seeing an aleph in one instance anyway, we are only able to conceptualise ("visualise", if you like) finity, a defined chunk of infinity. Thus, aleph 2 (or something comparable) will always appear to be larger than aleph 1, even though in infinity this may not be so.

    What about the concept of circular infinity? Since an infinite numerical set is always composed as an infinite arrangements of only 10 symbols, there's as sense of a constantly repeating "mantra" in different combinations. This helps somewhat to get your head around something generally conceptualised as linear.

    What do the mathematicians think of this?

    01 Jun 2006, 23:42

  5. There are lots of different mathematical conceptions of the 'size' of a set. The aleph notation always refers to cardinality. The idea of cardinality, as Thomas mentioned, is that two sets have the same cardinality if you pair them off one against another. Think of it like: suppose you have two bags of objects, and there are too many to count them all. You could just line them up, pairing one object in one bag against one object in another bag. You won't know how many there are in total, but you'll know that there are the same number in each bag. Similarly, for two mathematical sets, if you can find a correspondence between the elements of two sets they are defined to have the same cardinality. Thomas described a way of pairing off the set 1, 2, 3, … with the set 1.0, 1.1, 1.2, 1.3, … (the number n is paired with the number n/10), so the cardinality of these two sets is the same.

    But as I said, there isn't only one conception of size in maths. There's a clear sense in which the set 1.0, 1.1, 1.2, … has 10 times as many elements as the set 1, 2, 3, …. There's not a standard mathematical term for this sense that I know of, but certainly mathematicians wouldn't deny it. The difficult thing is to come up with a definition that works in all cases. It's "obvious" that there are 10 times as many elements of 1.0, 1.1, 1.2, … as 1, 2, 3, …, but think about this: the set of primes 2, 3, 5, 7, 11, … compared to the set 1, 2, 3, …. How much smaller is the former than the latter?

    There are ways to answer these sorts of questions mathematically, but not one that works universally. I think the idea you describe would be some sort of asymptotic relative density. You would say something like: if you look at all the numbers in the set less than 100, then there are 10 times as many in the one set as in the other set. The same is true if you look at all the numbers in the two sets less than 1000. Or 1000000. For the example I gave with prime numbers, this 'asymptotic relative frequency' gets smaller and smaller. It might be something like there are 15% as many primes less than 100 as integers (1, 2, 3, …). But only 8% as many less than 1000 (because primes get more and more rare as they get bigger and bigger). When you get up to 1000000 it might be only 1%. As you keep going the percentage gets smaller and smaller so the 'asymptotic relative frequency' of primes is 0%. This is one sort of mathematical answer to your question along the lines you mentioned. There are many others.

    I'm not sure I understand what you mean by your comments about circular infinity.

    02 Jun 2006, 01:03

  6. Oh, also about the alephs. The aleph notation refers to the cardinality of sets, as I mentioned. Aleph–0 is defined to be the smallest infinite cardinal number (which is the cardinality of the set 1, 2, 3, …). Aleph–1 is defined to be the next smallest infinite cardinal. It is bigger than aleph–0. The real numbers have cardinality 2aleph-0, also called c, which is definitely bigger than aleph-0, but it isn't known whether or not 2aleph-0=aleph-1 or not. (The really scary thing is that it has been proven to be unknowable, but that's another story. It's called the continuum hypothesis.) The 2 in 2aleph-0 is, as Thomas says, not important. It could be 3, 4 or any other number. It even turns out that aleph-0aleph-0=2aleph-0.

    02 Jun 2006, 01:09

  7. Circular infinity: the fact that only ten (or arguably even only two: 0 and 1) different symbols are needed to continue a sequence of numbers into infinity, this string of numbers has an essentially circular, or at least spiral–like character to it.

    It is 0 to 10 in new combinations, over and over again. I suppose original infinity, with continual totally original symbols entering the set, are not even possible. Or are these the "uncountable infinities" that you refer to, Tom?

    03 Jun 2006, 10:46

  8. Stephan

    Math can’t count. It is 0. That’s your infinity—the beginning.

    26 Jun 2011, 06:11


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