Can infinities have different sizes?

[JonF]

So the Real numbers, at least, are the ones you can count (tick, tick, tick - 1, 2, 3) and if the ticking doesn't end you have infinty?

I just can't think of a set of numbers that doesn't do this, even the Imaginary numbers, I suppose you can just tick tick tick the multiple of i (?)

I have no clue what this means. But, how about the set of all transcendental numbers?

the original question asked "If I count on my fingers the amount of real numbes between 1 and 2 OR if I count on my fingers the amount of integers between 1 and infinity do I count different amounts?"

This is how I’m going to interpret your question. I haven’t taken much more math than you, so I may be stating this badly (or wrong all together).


Let P be the set that contains all of the real numbers greater than 1 and less than 2.
Let Q be the set that contains all of the integers greater than 1.


Is either of these what you were trying to ask?

Are there more elements in P than Q?

Is there an operation that will, for every element in P generate a unique element in Q and generate all of the elements in Q?

 

[shrumeo]

Ok, folks, I've got it.

IT really is just a matter of semantics.

I'm sitting here using (my) the CONCEPT of infinity.

Yous math guys are calling sets of numbers "infinities."

Y'all are so used to it that you understand each other when you say "sizes of infinities" when to let it make sense to a layman like me, you should probably use the correct terminology which is "cardinality of sets."

To me, infinity is infinty forever and ever. :)

BUT, infinite sets can have different "sizes" or cardinalities.
This is obvious to me as well.
But, calling different sets of numbers different infinities is like saying a ray and a line are "different infinites" when to me they are different types of infinite objects.

So fergit it!

But you will be proud to know that I'm in a hurry because I'm on my way to a lecture by Brian Greene on string theory... :)

 

[Cognizant]

Voltaire postulated on this subject in Lettres Philosophiques.

"That there are infinite squares, infinite cubes, and infinites of infinites, all greater than one another, and the last but one of which is nothing in comparison of the last?

All these things, which at first appear to be the utmost excess of frenzy, are in reality an effort of the sublety and extent of the human mind, and the art of finding truths which till then had been unknown."

 

[matt grime]

I'm sitting here using (my) the CONCEPT of infinity.

and the penny drops! The difference being we have a well defined notion of cardinality

Yous math guys are calling sets of numbers "infinities."

No we are not. We are calling equivalence classes of sets cardinal numbers. Sort of the exact opposite of what you just said.

Y'all are so used to it that you understand each other when you say "sizes of infinities" when to let it make sense to a layman like me, you should probably use the correct terminology which is "cardinality of sets."

No, I think you'll find we took great care to explain to you that "infinity" and "size" are laymen terms that should be avoided, and instead you should talk mathematically to a mathematician.
We talked about countability, bijections, maps, sets, and so on. You were the one talking about counting sizes on your fingers.

To me, infinity is infinty forever and ever. :)

BUT, infinite sets can have different "sizes" or cardinalities.
This is obvious to me as well.
But, calling different sets of numbers different infinities is like saying a ray and a line are "different infinites" when to me they are different types of infinite objects.

No mathematician would call either of those sets an infinity, or a rype of infinity. We may say that the number of points in the set defined by a ray and a line are infinite. We would also say that they have the same cardinality, actually, so I don't think you do understand the concept of cardinality as it happens.

So fergit it!

But you will be proud to know that I'm in a hurry because I'm on my way to a lecture by Brian Greene on string theory... :)

Can't forget it if you don't learn.

 

[shrumeo]

these ideas are not easy. yoiu are in good company questinoing them. in his dialogues on two new sciences, galileo discusses the curious interplay between finite and infinite things.

if you take a finiute interval say of length one, and subdivide it as follows: first take half of it, length 1/2, then take half of what remains : length 1/4, etc... you can imagine subdividing a finite interval into an infinite number of pieces, of lengths

1/2, 1/4, 1/8, 1/16,...

hence if you add together all those lengths 1/2 + 1/4 + 1/8 +... you should get 1. the length of the original interval.

Is this a question about infinity or not?


the same question arises in asking why or whether .3333... = 1/3.

there are an infinite number of terms on the left and together they represent an infinite sum .3 + .03 + .003 +..., and the question is whether this infinite addition problem makes sense and equals 1/3.

try to get beyond the simplistic attitude that something is "either infinite or not". i.e. the equation .333... = 1/3 invovles a set of infinitely Many intervals whose total Length is finite.

this is a bit like the famous "hottentot" attitude toward numbers, either they are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or infinite. since then they run out of fingers and toes.

by the way it is not true that the reals are the set that can be counted off, tick tick tick; that's the integers that can be. the reals are much more extensive.

This sounds like a discussion of limits. Calc 101.

Did I say real numbers get ticked off? If I did I didn't mean to.
I'd say it's not the integers either but the Natural numbers that we use (subconsciously) to go tick tick tick.

what on Earth does "ticking" mean? Well done, once more you've introduced an undefined object into a discussion.

Ticking means putting your set in a one-to-one correspondence with the Natural numbers. You know, counting.

I know I know, not all sets are "countable."

You've not defined what it means to count an infinite number of objects, that is the whole issue here. The preCantor argument was that there was no meaningful interpretation to the notion of counting an infinite set.

I wonder how much "meaning" is being injected into counting infinite sets artificially. All we can really do about infinite sets (not infinities as they have been called in this thread and on websites that I have visited while discussing this thread call infinite sets) is extrapolate and assume.

Cantor did what a lot of mathematics is. He said, in effect "if we think about the finite case, and write a condition that is equivalent to two finite sets having the same size, and it doesn't refer to their finiteness, then we can apply it to infinite sets.''

Sounds like a hypothesis. But I'm sure it has been "proven."
Has anyone yet to count to infinity?

and once more we have an undefined term: 'counting', also 'same', we'll leave 'amount' alone.

You mean like plural "infinities?" I'll start leaving that one alone, then.

you don't see a diffference since yo'uve failed to fully articulate these ideas mathematically.

By taking the idea an expressing it in cryptic heiroglyphics?
Is that what you mean?

please, for pity's sake, read about this some more before posting more of this line of unfounded reasoning.

What I have read fails to satisfy me, so I argue it with people who think they know something and maybe they'll explain it to me so they can feel smart.

and the penny drops! The difference being we have a well defined notion of cardinality

That's a new expression for me (the penny thing).

We? Who is it that keeps saying infinity of infinties?

No we are not. We are calling equivalence classes of sets cardinal numbers. Sort of the exact opposite of what you just said.

The title of the thread is "more than one infinity." Right?
Did you clear up that with the poster or do you think this phrase means something?

No, I think you'll find we took great care to explain to you that "infinity" and "size" are laymen terms that should be avoided, and instead you should talk mathematically to a mathematician.
We talked about countability, bijections, maps, sets, and so on. You were the one talking about counting sizes on your fingers.

Infinity is a layman's term and should be avoided?
Maybe I should have avoided this thread altogether.

You can keep talking about maps and bijections if you want, but get the jargon straight so you don't confuse people who aren't on the same page.
I don't think others (not you, of course) should talk about plural infinities and types of infinities and degrees of infinity, when these have little meaning in the english language.

No mathematician would call either of those sets an infinity, or a rype of infinity. We may say that the number of points in the set defined by a ray and a line are infinite. We would also say that they have the same cardinality, actually, so I don't think you do understand the concept of cardinality as it happens.

Hmm, I'm the one that said people on this thread need to start using the phrase cardinality of infinite sets instead of sizes of infinities.

Can't forget it if you don't learn.

Yes, we do have problems with the language then.
Do you mean that YOU can't forget it until I learn?
Or are you saying that I can't forget something until I learn about it?
Or should I not forget until you learn something?


Anyway, it's such a little cosmic convergence.
Brian Greene gave his lecture and it wasn't what I expected.
I thought he would give a less dumbed down version of his book and TV show, but it was even more dumbed down. So if you watched the show or read the book, then you didn't learn anything. But maybe we should take a page from him. Relatively speaking there are very few people in each scientific discipline and most have very specific ideas and terminology. We can use these things when communicating to others in our field, but in public, when trying to explain them or discuss them with "lay" people then we shouldn't get so upset or insecure when they don't use our language. And if something should be taught it shouldn't be done bitterly, sarcastically, or with condescension.

 

[shrumeo]

So anyway, I was reading more about this.
Thanks for the advice!

:)

No, really, I came across some sets of numbers I have never heard of and can't really imagine.

The Hyperreals : http://www.daviddarling.info/encyclopedia/H/hyperreal_number.html
and the Surreals : http://www.daviddarling.info/encyclopedia/S/surreal_number.html

It's amazing and kind of incredible that there are numbers between zero and the next real number. I just wonder how they came across this and how many people in the world understand it enough to truly see the value or lack thereof. They sound invented, but do they have a use?

 

[Hurkyl]

For one, they're algebraically indistinguishable from the real numbers. In some sense, they're also analytically indistinguishable.

However, since the reals form a subset of the hyperreals, this allows one to do interesting things by comparing their analyses. (aka nonstandard analysis)

Surreal numbers have applications in game theory.

 

[mathwonk]

shrumeo:

"Maybe I should have avoided this thread altogether."

people are literally crying in their beer at this cruel threat.

 

[matt grime]

"The title of the thread is "more than one infinity." Right?
Did you clear up that with the poster or do you think this phrase means something?"

I think we cleared it up very well, and yes, that phrase means absolutely everything to the argument, which is essentially clarifying what on means by "different sizes of infinity"

 

[mathwonk]

After reading Matt's post, I am led to suggest summarizing the discussion as follows:

1) mathematicians look at this question of "size", following Cantor, as one of determining whether or not two sets have the same cardinality.

2) the definition of two sets having the same cardinality does not involve whether the sets are finite or not, it simply says they do iff there exists a bijection between them.

3) There exist two non finite sets, that fail to admit a bijection between them, namely the set of natural numbers, and the set of all subsets of the natural numbers.

4) Hence if you accept this formulation of the original problem, then it seems settled.