Why are infinites (or at least some of them) equal?

[CrystyB]

We have here a VERY nosy and curious being (namely Crazy Bob) which just asked "who sais infinites are equal?". I am afraid i would be too complex for a 12 years old genius, and i am no teacher either. Who'd give me a hand?

[Ghost Post]

Well, the quick answer is that they're not equal.

Cantor, if memory serves, was able to demonstrate that the infinity of irrationals is greater than the infinity of integers.

There are levels of infinity, Aleph-Null being the lowest level. Demonstration of some of the higher level ones becomes abstruse quickly.

[Ghost Post]

Infinities are equal if a one to one correspondence can be established, for example the number of integers equals the number of even integers because each member of one group has a corresponding member of the other, even though intuitively it would seem that there are more integers. Whew! that was a long sentence.

[Ghost Post]

I'd hope that this topic might include consideration of something that's been bugging me recently. There's a significant difference between *infinity* and *very large finite numbers* in our ways of looking at the known universe.

Even if the number of stars, or atoms, or elementary particles, is not infinite, it would take an inordinate amount of time to count them all. But since they are at least theoretically countable, isn't it a mistake to treat them as though they were indeed infinite?

The implications, in how we conduct our lives, are that there is only x-amount of material in this system, and that further implies (at least to me) that we need to consider the exhaustibility of material things. There's just x-amount we can waste or use.

There's a movie (maybe many, but not an infinite number) that pokes into these issues in a somewhat intelligent manner. We just saw it for the first time this week, although it's been out for quite a while.

Sam Waterston, Liv Ullmann, and John Heard in *Mindwalk* (1990), portray a politician, a physicist and a poet who delve into these and related issues.

While it's fun and tantalizing to investigate infinity in our minds, isn't it also valuable to cope with finitude?

[Ghost Post]

Yes, but it's like comparing a full eclipse and a partial eclipse. The difference is like day and night.

[Ghost Post]

this argument is really about definition of terms, if you can form a bijection (one-to-one corrospondance) between 2 infinite sets, then they would be considered as having the same magnitude of infinity.
but "size" in this discussion has nothing to do with the "size" comparing the values 6 and 6000.

for example, the set of all integers is equivilant in magnitude to the set of all positive integers.. although by standard notions of "size" this makes no sense.. as the positive integers are a subset of all integers.

daver: who considers the universe to have infinite material?
you are quite right in that there is really only finite.

but i will take you up on your argument about resource use...
what happened when we couldnt gather or hunt enough food? did we go hungry?
no, we learnt to farm.
what happened when there was not enough fresh water for people? did we die?
no, we learnt to process it.

do not forget that when you use "x-amount" of material it doesnt dissapear.. it only changes form... throughout history technology has made us able to turn "waste" into useful materials

[Andy]

Boom's point is well taken. We seem to have an intuitive understanding of what makes two finite quantities equal (there is a strict mathematical definition, and it turns out that equality is an example of an equivalence relation). We have no such intuitive understanding of equality of infinite quantities, so we relay solely on a formal definition - and the generally accepted definition is that two infinite sets are the same size if and only if there exists a one-to-one correspondence between the members of one and the members of the other. This definition doesn't require that every such correspondence be one-to-one, so we get some counterintuitive results.
For one example, there are the same number of positive even integers as there are of integers.
For another example, there are as many positive integers as there are positive rational numbers - even though there are infinitely many rational numbers between any two consecutive integers!

Note however that not all infinities are equal - e.g. there are more positive real numbers that there are positive integers. This means that any correspondence between the positive integers and the positive real numbers, even if it pairs up all the positive integers, must leave some real numbers unmatched.

[This message has been edited by Andy (edited 01-31-2000).]

[Ghost Post]

In answer to boom's question about who thinks things are infinite, I'd just say that there are many people whose knowledge of infinity is hampered by minimal understanding of the mathematical versions. Their ability to see "large numbers" as "infinite" leads some to think that there's no real limit to the universe or its contents.

Many of the New Age proposals for "infinite possibilities" and "infinite universes" seem to suffer from this problem. It seems to me that there are only x-number of distinct thoughts available. Some have already been thought, others are waiting to be thought, but there's some finite number of them, I suggest.

[CzarJ]

OK, Crysty, you obviously don't like me... and I would like to clear something up. I did not say "Who said infinities are equal?"
What I said was, "Who said the infinities were equal?" in response to an answer to a puzzle a while back, where someone's post, dealing with two seperate infinities, implied that they were equal. I'd get more specific if I remembered about it and it would make perfect sense. OK? The truth is, I already know most of this stuff, although, if you really enjoy discussing it, I'm all for it.

[mithrandir]

Just wondering if anyone can prove that the reals and integers can't be put in a 1-1 correspondence. I've got a proof I remember if you're interested, I'll post it in a couple days if no one puts one up.

[Ghost Post]

DaveR: i dont really think people believe that something big is something infinite.. its just a workable approximation.
like ignoring relativistic effects when designing a racing car...

anyone trying to work in the finite value of resources in the universe into some kind of long term prediction of things to come would be wasting their time, other variables are going to have so extremely larger effects on our future that it is irrelevant.

mith: that would be interesting.. i had an exam question where we proved that once.. but we were given some relationships between other sets to work with so it was quite easy, i have no idea here to begin with it

[Tom]

Erm .. I don't want to cause trouble, but there are actually 2 sorts of "numbers" (ie defined numbers, and we need to define them to talk about infinity). We have cardinals, when it is true that a 1-1 correspondence makes them equal, but also ordinals, when this is not enough. With the ordinals, the order is important (hence the name), so the first infinity, we call omega (we find it after 0,1,2,3,...) is not equal to omega + 1. This is because omage+1 has a specific "last bit" (the 1 added on), which omega alone does not. But, interestingly (well, to some..), 1+omega _does_ equal omega. So 1+omega is not equal to omega+1.

I'm guessing the proof mentioned above is Cantor's diagonal theorem?

Cheers,
Tom.

[Amy]

Wait, I didn't follow that. What is the definition for ordinals? I always thought ordinal numbers were "1st," "2nd," "3rd," etc. And then what is it you are defining as omega?

[Tom]

Yep, the ordinals are 1st, 2nd etc .. I just mean omega to be the first infinte one, ie just after all the finite ones.

More exactly (set-theoretically):

we define 0=empty set

n+1 = n union {n}

so any number is in fact the set of all the numbers before it (ie 3={0,1,2}. Then omega is the set of all the finite numbers, ie omega={1,2,3,4,...}

make sense?

omega is then not equal to omega+1, for exactly the reason that it is not of the form n union {n}, for some n.

Tom.

[Andy]

Mithrandir - yes, I remember such a proof - it's fairly obvious once you see it, although some people are philosohically opposed to reductio-ad-absurdum proofs. I also will allow more time for others to present such a proof; someone might have a different one.
Tom - if I remember correctly, Cantor's diagonal theorem offers a 1-1 correspondence between integers and rationals. ?

[mithrandir]

I'll go ahead and post the one I remember. To simplify, I will prove that the Natural numbers (which are the same amount as the integers) with the Reals over [0,1]. Now, let's assume I can have a 1-1 correspondence between the two sets. So, there is some function, f, such that each Natural number maps to one and only one Real in [0,1]. Now, we can list these:

f(1)=.a[1,1]a[1,2]a[1,3]...
f(2)=.a[2,1]a[2,2]a[2,3]...
f(3)=.a[3,1]a[3,2]a[3,3]...
.
.
.
and so on. Each a[m,n] is the nth digit in the decimal representation of f(m). Now, lets define a number T=.b[1]b[2]b[3]...
such that b[n]=any digit other than 0,9, or a[n,n]. Clearly, T can not be mapped by f(n). But, now we have defined T in [0,1], so that no Natual number maps to it, so f can not be 1-1.

[CrystyB]

Why do you take for granted that the UNIVERSE is finite? I was told (yes, a _very_ long time ago, but this is irrelevant) that it is infinite, and i can't remember anyone proving it is finite...


And another thing: I read in a book entitled "1, 2, 3... infinite" that there has not been found any set with cardinal larger than Alef-2 (the number of all curves, or if i'm right the number of all functions R->R). In your oppinion, is this true?

[Ghost Post]

Given an infinite set, S, of any cardinality, the set of all subsets of S is an infinite set of higher cardinality. This is easy to show. Take any mapping which maps members of S to subsets of S. Let's say that mapping if function f(x), where x is in S, and f(x) is a subset of S. Now consider the set T, a subset of S, defined as follows:

for all x in S, x is in T if and only if x is not in f(x)

T is not equal to f(x) for any x (because for any x, T contains x if and only if f(x) does not contain x)

So you can't map members of S onto the set of subsets of S, and so the set of subsets of S has greater cardinality than S.

[Ghost Post]

Here are the results of my research as an amateur astrophysicist. Please let me know if I've screwed something up (Very likely).

I think that nobody knows whether the universe in finite or infinite. We can only directly observe the part of the universe which is a neighborhood of the earth defined by a ball whose radius is the speed of light times the age of universe. Light from beyond this radius has not had time to reach our telescopes yet. There is no a priori reason to suspect that the universe is finite. In models of the original pre-inflationary big bang, the universe may be either finite (closed), or infinite (flat or open) in extent. In the flat and open cases, the universe is infinite in extent the instant that space-time comes into existence, i.e. space is instantly a 3 dimensional real or hyperbolic space.
(For more info, search the web!)

[Ghost Post]

Since irrationals are a subset of reals, then showing that the infinity of irrationals is larger than the infinity of integars will work. That isn't too hard, imagine an infinite list of irrationals, for example:

.1562737273...
.4353253432...
.8273642911...
.9283749939...
.2324584339...
.0092838421...
.
.
.
Even if this list is infinitely long, there are irrationals that arent in it. This can be shown by creating a new number that is obviously not in the list. Assuming two numbers are not equal when two digits in the same place column of those numbers are not equal. So the new number would have a different first digit from the first number in the list, by changing the .1 into a .2 . Then move to the 2nd number in the infinite list and increase it by 1 as well, so our new number is now .24, and so on... .248464... That number isnt in the list since it differs from each number in the list by at least one digit.

[CzarJ]

Wow. I really was a brat when I was twelve...

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[This message has been edited by CzarJ (edited 08-05-2001).]

[CrystyB]

No you weren't, only that you were a newcomer here, so you had no need to know that stuff. and now you realise how much you have to know if you hang out here ...

[CzarJ]

Well, if you say so, but I think I was just bein' a brat

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Why?