What are the different types of infinities?

[Avichal]

How can there be different types of infinities?
I just learned that cardinality of set of natural numbers, integers, prime numbers is aleph_o. Why are we replacing infinity with a number i.e. aleph_o.

And what further blows my mind is that cardinality of real numbers is also infinity but a different one - its is aleph₁.
How can there be different types of infinities?

Just as I thought I understood infinity I think I don't have a clue about it now.
Help me

 

[oli4]

Hi Avichal, you should lookup for transfinite numbers and continuum hypothesis, that should give you many useful links with several angles of attack so that one will talk to you better
In short, it is clear that there are different "infinities", as Cantor proved that there are clearly more real numbers than integers.
(with a nice enough definition of more that, for instance, you would think thee are more rational numbers than integers too, but that is not the case, thank's again Cantor
But then, dealing with the different infinities and categorizing them is much less clear (look for continuum hypothesis independence just to get a feel of it)
If you get hooked (I'm sure you will ;)) then I can recommend those books:
-> "The book of Numbers" (Conway, easy enough super fun)
-> "On numbers and games" (Conway again, but much harder)
-> "Set theory and the continuum hypothesis" (Cohen, very hard, but you can always 'ignore' (so to speak) the proofs and still understand what the theorems and their implications are about)

Cheers...

 

[D H]

The infinities that arise in analysis are different beasts than the the infinities that arise in set theory. Your problem is with the infinities in set theory. So let's look at those.

The idea of infinities in set theory is an attempt to measure the cardinality or, or number in a set -- even when it's infinitely large. The sets {1,2,3}, {2,3,4}, and {2,4,6} all have three elements. One way is just to count elements. That works fine for finite sets, not so well for infinite sets.

Another approach is to try to establish a one-to-one mapping from one set to another. If you can, the sets have the same cardinality. This concept does extend to infinite sets. For example, the sets {0,1,2,...} (the natural numbers) and {1,2,3,...} (the counting numbers) are of the same cardinality because it's easy to make a one to one mapping from one set to another. Not quite so obvious, the set of all even integers and the set of all integers are also of the same cardinality because there's a one to one mapping that maps each element in one set to exactly one element in the other set. That infinite number of odd integers that we skipped over don't matter.

With finite sets, the cardinality of a proper subset B of a set A is always less than that of the cardinality of set A. That's not the case with infinite sets. The counting numbers are a proper subset of the natural numbers, yet they both have the same cardinality. The even integers are a proper subset of all the integers, and yet here too both sets have the same cardinality. That's perhaps the first stumbling block in understanding the cardinality of infinitely large sets.

This concept of creating a one to one mapping also works with the rationals. It's a bit hairier, but it is still possible to create a one to one mapping between the integers and the rationals. Another surprising result: It's possible to create a one to one mapping between the integers and the rationals between zero and one. So another twist you have to get your mind around.

How about the real numbers? You can't make such a one to one mapping with the integers. You can't even make a mapping between the reals between 0 and 1 and the natural numbers. Assume you can make such a mapping. Use base 10, or base 2, or any other base, to represent those real numbers. A mapping exists by assumption, so we have a first real number, a second real number, and so on. Now let's make a new number by picking as the first digit something other than the first digit in our first real, picking as the second digit something other than the second digit in the second real number, and so on. This number is different than the first real, and the second, and the third, and so on, by construction. But it's still a real number and it isn't in our one to one mapping. The assumption fails; the reals are "bigger" than the integers. This is Cantor's diagonalisation argument. You can keep going like this forever. There are sets that are bigger than the reals, sets bigger than that, and so on.

 

[Mentallic]

How can there be different types of infinities?
I just learned that cardinality of set of natural numbers, integers, prime numbers is aleph_o. Why are we replacing infinity with a number i.e. aleph_o.

And what further blows my mind is that cardinality of real numbers is also infinity but a different one - its is aleph₁.
How can there be different types of infinities?

Just as I thought I understood infinity I think I don't have a clue about it now.
Help me

And what's even more mind-blowing is that you can list all the integers in an infinitely big book, as well as the rationals, but you can't list the reals. There are just too damn many of them to list in an infinitely big book!

 

[Avichal]

Its still not clear - infinity is just a concept to represent something which is limitless.
Natural numbers, integers, prime numbers, rational numbers are all limitless. There is no bound on their limits. So they are all infinite. Its hard to treat each infinity as different.
How do I visualize each infinity as different?

 

[D H]

To add to what Mentallic just said,

Even more mind-blowing (to me at least), you can't even find a way to compute all of the reals. It's not just a few special cases that are missed. Almost all of the reals are not computable. Just as one can map the rationals to the integers, one can map the set of all expressions that can be formed from a finite number of symbols to the integers. In other words, the set of all possible mathematical expressions is countable, which in turn means that the set of real numbers that we can represent by some algorithm is a countable set. The reals are uncountable. The best we can ever do is represent an infinitesimally tiny fraction of them.

 

[Hurkyl]

That argument is a little slippery, though: it has issues similar to those involved in Skolem's paradox.

 

[Diffy]

Perhaps the FAQs regarding infinity would help? Not sure if you have read through them or not

https://www.physicsforums.com/showthread.php?t=508840

 

[arildno]

Its still not clear - infinity is just a concept to represent something which is limitless.
Natural numbers, integers, prime numbers, rational numbers are all limitless. There is no bound on their limits. So they are all infinite. Its hard to treat each infinity as different.
How do I visualize each infinity as different?

Basically, you CAN'T visualize it.

However, for some infinities, you can set up a scheme that can COUNT them in principle.
For other infinities, you can't set up a scheme to count them with the counting numbers, not even in principle.

 

[pwsnafu]

That argument is a little slippery, though: it has issues similar to those involved in Skolem's paradox.

To be more precise, Skolem's paradox gives us a concrete limit on the power of first order logic. FOL cannot properly describe the infinite domain, we need second order logic for that.

 

[SteveL27]

Its still not clear - infinity is just a concept to represent something which is limitless.
Natural numbers, integers, prime numbers, rational numbers are all limitless. There is no bound on their limits. So they are all infinite. Its hard to treat each infinity as different.
How do I visualize each infinity as different?

It's easy to learn to do this.

The trick is to forget the philosophy and learn the math.

We may or may not have a sense of the nature of philosophical infinity -- the cosmos, the universe and everything in it, the limitlessness of possibility, etc etc etc.

Forget all that. Just put it aside. Save it for later if you enjoy contemplating it. But it has nothing to do with the mathematical study of infinity.

For the moment, just study the math.

There are sets that can be put into 1-1 correspondence with the natural numbers. There are sets that can't be put into 1-1 correspondence with the natural numbers

After you've become familiar with the basic proofs -- by familiar, I mean you work through them over and over till you can reproduce them at will and you begin to understand their content -- you'll be able to visualize the different infinities. You'll develop intuition about when a set is likely to be countable or uncountable.

Like anything else, the more you do it, the better you get at it.

It's a reasonable philosophical position that set theory does not capture everything about the philosophical infinity. After all, if we can talk about it, it must not really be the real infinity! As Lao Tzu said, The Tao that can be spoken of is not the ineffable Tao. This is pretty much what he was getting at.

Cantor himself had many mystical views and tried to relate them to his discoveries in set theory. Nowadays Cantor's set theory is ubiquitous in math; but his philosophy's all but ignored and forgotten.

I hope some of this helps. You can study set theory and get a very good sense of the relative sizes of infinite sets. Whether that satisfies one's personal intuition about the philosophical infinity is a different question. What tends to happen is that mathematicians don't care much about philosophical issues; but philosophers do. Most mathematicians don't worry too much about philosophy. But there are some serious philosophers of math who do give these matters some thought.

There's a very excellent book that discusses the mathematics of infinity in the context of the quest for the "actual" infinite, whatever that means. It's called Infinity and the Mind by Rudy Rucker.

https://www.amazon.com/dp/0691001723/?tag=pfamazon01-20

One time I was reading it before bedtime and fell asleep half-dreaming of the tower of countable ordinals. It was a disconcerting and dizzying experience.

The ordinals are much stranger than the cardinals in fact. The cardinals are the Alephs. But even with a set of a given cardinality, such as a countable set, there are many ways of ordering the set; and then ordering all the different orders. I find the ordinals much more mind-boggling than the cardinals.

 

[Hurkyl]

It's a reasonable philosophical position that set theory does not capture everything about the philosophical infinity.

Honestly, assuming you meant to say "mathematics" rather than "set theory", I've been given the impression of the exact opposite: that the rigorous mathematical treatment of issues of the infinite have been so successful that the subject is now considered to be a wholly mathematical topic.

 

[pwsnafu]

Honestly, assuming you meant to say "mathematics" rather than "set theory", I've been given the impression of the exact opposite: that the rigorous mathematical treatment of issues of the infinite have been so successful that the subject is now considered to be a wholly mathematical topic.

This is my impression as well. You talk to philosophy students who make arguments like "humans can understand infinity, machines can't, so AI can never be outclass humans", and just roll your eyes. Philosophical infinity is pretty much useless for logic purposes now.

Edit: I just came back from the bakery and just realized something. If anything our intuition is closer to the ordinals than "one infinity to rule them all". Just look at kids, especially grade 3~5 range. Ask them what is the largest number, you'll get answers like "million" or "billion", but one student will say "infinity". Sometimes someone else will say "infinity plus one". This is exactly the ordinal numbers. Kids don't have a problem with numbers larger than infinity or towers of infinity. If you ask them "what is infinity minus infinity" it's an emphatic zero. Only when you explain Hilbert's Hotel do they stop go "huh?"

 

[Bacle2]

DH: Small nitpick: bijection instead of injections, or maybe you meant 2-way

injections (Rick-Schroeder-Bernstein).


Something else to consider re the different levels of infinity is that , for any set S,

|P(S)|>|S| , i.e., the powerset P(S) of any set S cannot be put in 1-1 correspondence

with S itself. That gives you a chain of infinities, and shows that there is no set of

all sets --if there was one, it would be strictly contained in its powerset.

 

[Hurkyl]

This is my impression as well. You talk to philosophy students who make arguments like "humans can understand infinity, machines can't, so AI can never be outclass humans", and just roll your eyes. Philosophical infinity is pretty much useless for logic purposes now.

Edit: I just came back from the bakery and just realized something. If anything our intuition is closer to the ordinals than "one infinity to rule them all". Just look at kids, especially grade 3~5 range. Ask them what is the largest number, you'll get answers like "million" or "billion", but one student will say "infinity". Sometimes someone else will say "infinity plus one". This is exactly the ordinal numbers. Kids don't have a problem with numbers larger than infinity or towers of infinity. If you ask them "what is infinity minus infinity" it's an emphatic zero. Only when you explain Hilbert's Hotel do they stop go "huh?"

I'm skeptical: it's far more likely they're just extrapolating what they know about natural number arithmetic. Surely they will also tell you that "one plus infinity" is bigger than infinity, and that "infinity minus one" is the number that comes just before infinity.

Such things would be valid for non-standard models of arithmetic (e.g. hypernatural numbers) -- except that it still would be inappropriate to use the word "infinity" as if it referred to a number in the system.

 

[pwsnafu]

I'm skeptical: it's far more likely they're just extrapolating what they know about natural number arithmetic. Surely they will also tell you that "one plus infinity" is bigger than infinity, and that "infinity minus one" is the number that comes just before infinity.

Agree completely. Kids don't have a good sense of scale, so the difference between a million and a billion is a letter. I remember when I was in primary school I had difficulty visualizing how long a century actually was. To kids "infinity" is just another number. Similarly, kids don't really see the differences between naturals and rationals either. They are all just "numbers".

Such things would be valid for non-standard models of arithmetic (e.g. hypernatural numbers) -- except that it still would be inappropriate to use the word "infinity" as if it referred to a number in the system.

Yup. Infinity is ugly word.

 

[Deedlit]

That argument is a little slippery, though: it has issues similar to those involved in Skolem's paradox.

Are you talking about D.H. saying that almost all reals are uncomputable, or that the set of mathematical expressions is countable? I'm not sure about the latter, but I thought that it was uncontroversial that the computable numbers were a countable set, and hence nearly all real numbers were uncomputable.