Could my infinity be bigger than your infinity?

Factual Questions
1

Please indulge me to preface my question with a relevant but somewhat showoffy cute kid story. When I put my 5 and 8 year old kids to bed each night, I always ask them how many times they want me to check on them. Lately they have been saying “infinity times” so I respond by saying, “OK, first I’ll check on one of you infinity times, and when I’m done, then I’ll check on the other.” My daughter (smart as a whip) shot back “Why don’t you just alternate who you check on each time you come into the room so that you can check on both of us infinity times?”
That got me to wondering whether there are different “qualities” or “densities” of infinity.
First example: There are an infinite number of positive multiples of 10, and there are also an infinite number of positive integers. But doesn’t there have to be ten times as many positive integers as there are positive multiples of 10? Woudn’t one of these infinite number spaces contain ten times as many numbers as the other infinite number space?
Second example: Imagine a hypothetical infinite universe with, say, an average denisity of one hydrogen atom per cubic meter, and another universe with a density of ten hydrogen atoms per cubic meter. Wouldn’t one of these infinite universes contain ten times as many atoms as the other, even though both contain an infinite number of atoms?
You guys are my only hope. The she-Sput (ever the engineer) dismissed my question by just saying “Infinity is infinity; I don’t know from this other stuff you’re babbling about.” :frowning:

Help.

2

No… there is exactly the same amount of positive integers as there are of positive multiples of 10…

If you were to say that one was less, then you would be putting a finite limit on it, and then it wouldn’t be infinity.

3

Yes, there are different infinities. Here’s a webpage with links to a number of other webpages which discuss this matter:

http://www.math.fau.edu/Richman/Ideas/infinity.htm

This has also been discussed several times on the SDMB. Do a search and you’ll find these threads.

4

“Infinite” is not a number. Try substituting the term “not definable” and then you will see that your question does not make sense. “Infinite” is a concept which basically means a number that cannot be defined. One cannot define the number of positive integers. One cannot define the number of multiples of ten.

Your wife is right. One can only say that there would be ten times as many hydrogen atoms in one universe than another if one defines how big the universe is. But you say each universe is infinite. That is, it’s bounds are not definable. So you can’t say how many hydrogen atoms there are in one or the other. They are uncountable. They are without end.

Imagine two cars. One is driving at 1 mph and one at 10 mph. They keep driving forever. Which one will drive the furthest? The answer is that you can’t say. They will both go an infinite distance, that is to say, a distance that cannot be defined. One can only say that at any given time, one will have gone 10 times as far as the other. Without specifying that time, the question of which will go the furthest, is without meaning.

Likewise for your two examples.

5

Yes, roughly speaking, if you took the number of integers, that’s the smallest infinity, in Cantorian Set Theory. It’s called Aleph-Null (see Wendell Wagner’s link). But since it’s the smallest infinity, then Aleph-Null divided by 10, which is still infinite, must be the same number. So, the infinities in your example are the same. 10 × Aleph-Null = Aleph-Null.

However, there are bigger infinities. For instance, the number of points on a line. This turns out to be more like 2[sup]Aleph-Null[/sup], which is not Aleph-Null. This is called Aleph-One.

One problem I think is that Algebra does recognize any infinite entities. (Someone can correct me if I’m wrong about this.) That is, there is no such thing as infinity in Algebra. This is what Princhester is speaking of. There is such a thing as infinity in Calculus, and in Set Theory, but not in Algebra. So applying these Algebraic concepts like multiplication to infinities is really mixing ideas.

6

I say we ban questions on infinity from the SDMB.
I’ve still got a headache from the “1=inf” thread. I finally got that one, so leave me alone. :wink:
Peace,
mangeorge

7

I don’t know if anyone brought this up before, but on NPR, I heard a polymath describe the analogy of the Infinite Hotel. A man comes to the Infinite Hotel, which has an infinite number of rooms, and asks for a room. The desk clerk says, “I’m sorry, but we have an infinite number of guests tonight, so we’re all booked up.” The man says, “have the guest in room 1 move to room 2; have the guest in room 2 move to room 3, and so on, and I’ll stay in room 1.”
I worked in a hotel for five years, so my addition to this analogy was, “and the desk clerk said, ‘certainly. You can have room 1 just as soon as all of our present guests have been taken care of.’”

8

This is the old continuum hypothesis rearing it’s ugly head again. While it’s true that 2[sup]Aleph-Null[/sup] is the number of points on a line, and that this is bigger than Aleph-Null, it’s not necessarily true that 2[sup]Aleph-Null[/sup]=Aleph-One. The continuum hypothesis says that they are the same, but with the standard ZFC axioms, it’s actually undecidable. It is consistent that 2[sup]Aleph-Null[/sup]=Aleph-Two, or Aleph-Three, or just about any other cardinal (actually, it is consistent for 2[sup]Aleph-Null[/sup] to be any cardinal with cofinality greater than Aleph-Null, I believe).

While things in algebra do have a certain “finiteness” about them (being that much of algebra is based on binary operations, you’re only dealing with finitely many elements at a time), you can deal with infinity in algebra, as well. For example, the arithmetic (addition and multiplication) of cardinals and ordinals.

9

That would be Hilbert’s Hotel.

I’m in the middle of “Infinity and the Mind” by Rudy Rucker - it deals with exactly this type of subject. Most of it is accessible though some requires rather deep math. I believe it was recommended on this very bulletin board in a similar thread a while back.

10

If it is, I’ll buy a Lexus!!!

Ba-da-bing!!!

Thank you, I’m here all week…

11

Thanks for all the replies - I have and will continue to check out the webpages recommended by Wendell. Though I haven’t done any Serious Math in a number of years, I’m not sure I totally agree with Princhester’s “infinity= undefined” concept. It seems that although infinity is not a number, it has certain qualities that make it useful in computation, such as summing a series as one of the variables approaches infinity. Back to the first anecdote, if I were to check on my son once, then my daughter ten times, then my son once, ad infinitum, as the number of cycles approaches infinity, wouldn’t the total number of times I check on my daughter approach ten times the total number I check on my son?

12

Yeah, that’s Calculus. Cabbage was correct in pointing out that Algebra can be done on cardinals and ordinals, but I don’t think that you can treat just plain “infinity” as you would another number. (Is that right, Cabbage, or am I still not getting it?) Although, oddly enough, my calculator has an infinity button. Let me see the results of some operations. Apparently…

infinity + 1 = infinity
infinity + infinity = infinity
2[sup]infinity[/sup] = infinity
infinity[sup]infinty[/sup] = infinity
infinity / infinity = undef
infinity - infinity = undef
sqrt(infinity) = infinity

Well I guess that clears things up. Thank you, Texas Instruments. :rolleyes:

13

I half-recall from a college math course the concept of “cardinality”. Perhaps someone with much better recollection of what that exactly is, and how it relates to infinity, can explain. My first fuzzy recollection is that it does somehow describe different types of infinities, but that might be very wrong.

14

You can tell that both your examples are the same infinity because you can set up a one-to-one correspondence between them. One pairs with ten, two pairs with twenty, three pairs with thirty…ten pairs with one hundred, and so on. Each number in one set has exactly one corresponding number in the other, and you will never run out and they will all have a match. You can tell that the number of points on a line has a higher cardinality (is a “bigger” infinity) because you cannot do this one-to-one correspondence with integers. Match one to one. Now what do you match two with? 1.1? 1.01? 1.00001? You can’t define a way to match them all up so that no points are missed. This is because not only are there an infinite number of points, but there are an infinite number of points between any other two points.

At least, I think this is how it works. Hopefully Cabbage will correct anything I’ve screwed up royally.

-b

15

There is more than one concept of infinity.

In calculus, infinity is not a number, but a formal symbol. That is you cannot use infinity in an expression. Calculus was originally formulated with infinitesimals (called fluxions) and infinities. It was thought at the time that actual infinities and infinitesimals were not consistent. Bishop Berkley (as I recall) referred to fluxions as “ghosts of departed quantities”. So actual infinities and infinitesimals were replaced by the epsilon-delta definitions that we all know and love. However, since then, systems of numbers that include infinities and infinitesimals (called non-standard analysis) have been shown to be consistent (or at least as consistent as standard analysis).

The transfinite ordinals and cardinals are numbers. They can be used in expressions. Cardinals are the counting numbers. As in: There are 12 eggs in a carton. There are aleph-null integers. There are aleph-null integral multiples of 10. Two sets have the same cardinality if it is possible to pair off the elements of the two sets. One characteristic of an infinite set is that you can find a proper subset with the cardinality of the full set. The power set of A (the set of all subsets) is always larger than A.

John Conway came up with a way to imbed the real numbers in a Field (Field as opposed to field because it is not a set, but a proper Class). Besides the reals, this Field includes infinitesimals and infinities. You even have such numbers as sqrt(aleph-null).

And of course, we have the Texas Instruments concept of infinity.

No. You would visit each aleph-null times. Aleph-null times 10 (or any non-zero finite number) is aleph-null. When you multiply or add two cardinals, if at least one of the terms is infinite, the sum and the product are both equal to the larger of the two.

You are correct that the set of reals has a larger cardinality than the integers, but not for the reason you gave. The rational numbers can be put into a one-to-one correspondence with the integers. I’d give Cantor’s diagonal proof, but this post is already long enough.

16

I know that I’m somewhat confused now, so I’m worried about the OP at this point. :wink: Here’s one thing that may be easy to understand. Any infinite set that you can list must have a cardinality of Aleph-Null. So for instance,

the positive integers: { 1, 2, 3, 4, 5, … }
the integers: { 0, 1, -1, 2, -2, 3, -3, … }
the prime numbers: { 2, 3, 5, 7, 11, … }
the multiples of 10: { 10, 20, 30, 40, 50, … }
the multiples of one-tenth: { 0.1, 0.2, 0.3, 0.4, 0.5, … }
and even the rationals: { 1/1, 2/1, 1/2, 3/1, 1/3, … }

all have cardinality Aleph-Null. So even though we intuitively think of some of these sets as being bigger than others, they’re all the same “size”.

DrMatrix, what’s Aleph-Null - Aleph-Null?

More fun facts from TI: infinity[sup]1/infinity[/sup] = 1.

17

I’m not at all sure, but I don’t think there’s any consistent way to define that. The difference of two countable sets may be countable (Q - N), finite but non-empty ((N U {-1}) - N), or empty (N - N). Any value for aleph[sub]0[/sub] - aleph[sub]0[/sub] would have to be consistent with all of these, if the infinite case is analogous to the finite case.

18

Subtraction is defined in terms of addition.
a - b is the number c that satisfies: a = b + c. Aleph-null = aleph-null + c is true for any finite c (including zero) and for c = aleph-null. Aleph-null - aleph-null is not well-defined. (Or on preview, what ultrafilter said.)

The set of real numbers cannot be put into a 1-to-1 correspondence with the integers. Nor can any set be put into a 1-to-1 correspondence with its power set. The power set is always larger than the set. Given any set, you can construct a set with a larger cardinality, by using the power set.

19

I think I can wrap my decaying mind around this as an answer to the OP that a non-mathematician can comprehend. And I think I get the difference between Aleph-Null and the “continuum” type of infinity (unclear to me whether that’s called Aleph-One or not). So there are different types of infinity, just not along the lines of my original question.

How in the world was this determined? Please pardon me if this is a dumb question, but would this mean a finite number system that ended at, say, 8, would only have 256 (2[SUP]8[/SUP]) points on a line?

gotta get me one of them there TI adding machines…

20

2[sup]Aleph-Null[/sup] is the size of the powerset of the naturals. So the continuum hypothesis is basically asserting that the reals are the same size as the powerset of the naturals. There is no good analogy with the finite case.