The True Infinity

Factual Questions
1

It is very possible to have an infinite set that doesn’t contain everything. For example, “all the integers above one-billion” is an infinite set, even though it’s missing possible numbers.

Then, I learn that there are infinities above infinities, like aleph-one, aleph-two, etc., and that the infinity I knew all my life was just Aleph-null.

So, my question is: Is there a number that is the True Infinity, that contains everything, and leaves nothing out?

My guess is that Aleph-infinity would: an infinity of infinity of infinity… ad infinitum. :slight_smile:

2

Infinity is not a number, and can’t be thought of as such. The example you gave, the set of integers above one billion vs. the set of all integers, are both infinate, because they go on forever. Something that is infinate does not have to encompass everything, though. It just has to not end.

I dunno anything about this aleph stuff, though.

3

No, there is no set that contains everything. For any set S, the powerset of S (the set of all subsets of S) is of bigger cardinality. But if a set S contains all the members of a set T, then S is of no smaller cardinality than T. So the universal set is at once smaller and no smaller than its powerset, and there is no universal set.

There are also infinite cardinals that are bigger than any aleph, but that’s a technical matter.

4

From Beginnings of set theory

5

Über-infinity

But wait a minute, what about Über-infinity + 1? Crap!

Your question is one more of semantics than the concept of infinity. Listen to Friedo.

6

From http://www2b.abc.net.au/science/k2/stn/archive2001/posts/March/topic268007.shtm

Ahhh - infinity - a concept that never ends.

7

As you may have guessed by now, the Alephs keep on going even after the subscript has become infinite. So you can have Aleph_Aleph_1, and so on. A good book for an account of the infinite cardinals and ordinals, together with some speculation on their metaphysical significance, can be found in Rudy Rucker’s Infinity and the Mind

8

lexi quotes the following from a website:

> Mathematicians divide infinite sets into two categories,
> countable and uncountable sets.

Well, no, not really. Mathematicians say that there are many kinds of infinity. The smallest of these is Aleph Null, which is a countable infinity. All infinities greater than that are uncountable infinities.

Mike38722 writes:

> My guess is that Aleph-infinity would: an infinity of infinity of
> infinity… ad infinitum.

Well, no, not really. In so far as I can interpret this, this is not Aleph Infinity, but just Aleph One (the infinity which is the number of the power set of Aleph Null). I will assume that the “infinity” that you are using in your definition is Aleph Null. So your definition means Aleph Null times Aleph Null times Aleph Null times . . ., etc. Now, if there had only been a finite number of Aleph Nulls being multiplied together, the product would still be Aleph Null. But since you want to know what the product of Aleph Null multiplied times itself Aleph Null times is, that’s Aleph Null to the power of Aleph Null, and that equals Aleph One.

9

Ok. I was looking forward to this fall semester when I will be taking chem 1 and college algebra. I am now scared. I know you are going further than just plain college algebra, but I also have calculus and trig ahead of me.

I know nothing.
The only calculus I know is the calculi that can grow in various parts of your body.
Cathy:eek:

10

UCOStudent,

You’re not going to learn about the different kinds of infinity in your class this fall. It’s the sort of thing not taught even to undergraduate math majors until at least their junior year in college. Usually it’s taught in a course called “Set Theory”.

11

Thank goodness. But I’m still coming back here when I get stumped.
Should I worry about the Plane Trig and Quantitative Analysis? Or general physics?
Cathy

12

Wait, I’m very confused. How can there be “more” of one infinity than another infinity? They are both unending, and therefore comparisons are impossible, right?
Also, to me, it would seem redundant to take an Aleph-infinity, an infinite amount of infinities? What? It was already infinite to begin with! It seems you can’t have anything more than infinity, because by definition it is already boundless. Adding anything to it still brings you to an answer of infinity.
I must be missing something. My high-school calc. course would never cover something like this…and I figure just about all of you know more about this than I do. Could someone help?

13

Look up Cantor’s diagonal proof. I’m sure there are good explanations on the web, I’ll take a quick crack at it.

You probably know that there is an infinity of integers, and an infinity of real numbers between 0 and 1. So let’s say you try to count them (I’m putting brackets around special digits):

1 0.[1]234552358…
2 0.2[8]3768295…
3 0.28[4]5638387…
4 0.847[8]4847…

and so on. Even if you list goes on forever, Cantor showed that you can always use this list to generate a real number that is not on the list! Just do it by going along the “diagonal” digits after the decimal, making sure you chose a digit that is different from the corresponding one on the list, in this example I could use

0.2959…

and keep going. A number constructed this way will be different from every one on your list - proving that the real numbers are uncountable, and that there must be “more” real numbers than integers. (the math experts here may point out that I’ve skipped a few details, but I hope I gave the proper flavor of the proof).

14

What’s missing from here is what it means for one infinity to be “more” than another infinity. The trick is that two sets are the same “size” if you can match each element in one set with an element in the other set. When we say that there are 3 letters in the word “dog” it is because we can match the letters with the numbers in the set {1,2,3} so d goes with 1, o with 2 and g with 3.

If 2 infinite sets are the same size, it should be possible to match all the elements in one set with the elements in the other. Even though it seems like there are “more” positive integers (1,2,3…) than positive even integers (2,4,6…) they are really the same size because we can pair up the positive even integers y with the positive integers x via the equation y=2x. Cantor showed that you can do the same thing, in a more complicated way, with the integers and the rational numbers (p/q, where p & q are integers), so they’re all the same order of infinity (countable infinity or aleph 0).

Cantor’s diagonalization proof, given by kellymccauley, shows that you can’t do this with the integers and the real numbers, so the real numbers have to be a higher order of infinity. (It turns out you can’t prove that this is aleph 1, but it makes sense to define it to be.)

15

It’s been a while since I studied set theory, but that doesn’t sound right to me. I thought there was an aleph for every cardinal. (Do I have to assume the Axiom of Choice, maybe?)

Two sets are the same size if there is a function from one set to the other that is one-to-one and onto. As kellymccauley pointed out, Cantor’s diagonal proof shows that any mapping from the integers to the interval [0,1] is not onto. There is an elegant diagonal proof that a set is always smaller than its power set (the set of all its subsets).

Assume that there is a mapping f from A to P(A) that is onto. For all a in A, f(a) is an element of P(A). Or f(a) is a subset of A. Consider B = {a: a is not an element of f(a)}. S is a subset of A, and so an element of P(A). Since we assume f is onto, there is a b in A such that f(b) = B. Now b cannot be an element of f(b) by definition of B. But if b is not an element of f(b), then it is in B. We have a contradiction, so our assumption that there is an onto mapping is false.

I love that proof.

16

Ok. I bite. What’s an aleph?:confused:

17

Just to play devil’s advocate here… (and be really annoying in my ignorance)
There is no such thing as infinity. Yes, I understand that you have a concept of endless sets, but no such thing exists, either in reality or in a concrete concept, merely your description of the concept. After all, the biggest number you can think of, plus one ad infinitum (or however you want to define it) is still just a function that cannot be carried out, except by the self defining rules.

Take for example the oft postulated infinite monkeys typing on infinite typewriters. It is oft said that eventually they will come up with all of Shakespeare’s plays. As I understand the concept of infinity, this isn’t correct. They will come up with all of Shakespeare’s plays, and every other work of literature and every possible bit of gibberish in about as long as it would take them to perfectly pound out the text without typoes the first time around. (Phone books and IRS regulations take longer because they are longer works.) But wait, this is infinity, that’s not all! Since this is infinity, that has no end, you could take every hundredth monkey and set them aside, and have 99% of your monkeys throwing monkey poo at each other and 1% still complete the same typing task, with the same results, because you still have an infinite number of monkeys working. You still get infinite copies of each great work and each piece of gibberish. But wait, it gets more absurd. You can come infinitely close to 100% of the monkeys throwing poo at each other and still have the same result, as long as it only approaches 100%, but never reaches 100%. After all, such a subset is still an infinity, but an infinitely small infinity. Yet since it is an infinity, the result is the same as for the largest infinity for our very pratical purpose of making an infinite number of copies of all works of literature and gibberish.

But wait! Ever wonder why mathmaticians use monkeys typing on typewriters coming up with literature? I’ve a better illustration.

An infinite number of monkeys with an infinite number of pencils and notepads randomly scribbling come up with all possible mathematical proofs in just a few hours depending on the length of the proof, laws of physics, grocery lists, etc. Remember, an infinite number of monkeys randomly solve each currently unsolved problem and at the same time create every as yet unknown problem, and solve them before they are finished being defined.

So remember, everytime you can’t solve a problem, remember, there an infinite number of monkeys can do better than you without trying an infinitismal fraction of how hard you are. Idiot.

I’ve waited years to actually put type that out. I just hope some part-time poo throwing monkey somewhere didn’t beat me to it.

18

Aleph is the first letter of the Hebrew alphabet.

19

Thanks. Now if I could just understand the rest of it. Interesting, even if I don’t understand it.
And I like the monkey poo concept. Sounds a lot like an Alzheimer’s ward I once worked on.
C

20

I just thought I’d mention that you can get an aleph in the symbol font. The keystroke combination on my machine is Alt+0192

entering this: **[symbol]À[/symbol][**sub]0[/sub], **[symbol]À[/symbol][**sub]1[/sub]

yields this: [symbol]À[/symbol][sub]0[/sub], [symbol]À[/symbol][sub]1[/sub]