Can infinities have different sizes?

[jcsd]

well to be fair mathworld describe infinity as an 'unbounded quantity', though waht's a quantity. The best way (though I'm prepeared to be contradicted) to think of infinity in terms of a 'number-like entity' (i.e. soemthing that we put were we usually put a number such as defining intervals in the reals) is to think of it in terms of one of the ordered sets (e.g. the extended reals, extended complex planes) that contain it. So for example we define the interval [0,infty) which is an interval in the reals, by using the order relation of the affinely extended reals.

 

[Hurkyl]

The main thing about the various ways to define the word "infinite" is this: when we say X is infinite, we mean that in some sense, the magnitude of X is greater than that of any integer.

It is much rarer to talk about "infinity". Infinite things are rarely called infinity.

----------------------------------------------------------

The main point to remember about the extended reals is that the two points at infinity (+∞ and -∞) are only expected to behave properly with respect to the ordering.

Any other properties they have are a nice bonus. For example, the + operator can be extended "to infinity" by continuity -- specifically, arithmetic has nothing to do with this extension.

You can do the same with the * operator... but one of the places to which * cannot be extended is 0 * ∞.

-----------------------------------------------------------

There are arithmetic methods that permit infinite numbers that behave properly with respect to arithmetic. But, in all of those cases, 0*x = 0, even for infinite x. (Why? Because they behave properly with respect to arithmetic. )

-----------------------------------------------------------

The "infinities" of set theory (that is, the infinite cardinal and ordinal numbers) are, again, simply a type of ordering, which are based on functions between sets. (order preserving functions, for the ordinal numbers) There are ways to define arithmetic of these things by analogy, but it's very strange. For instance, if a and b are infinite cardinals, a + b is simply their maximum. For ordinals, a + b is usually different than b + a.

0 * a = 0 for any cardinal, though.

 

[matt grime]

ok geniuses, explain it to me

why isn't infinity a quantity and 2 is?

The problem is that you didnt' bother to explain what you meant by quantity. You said in effect that there is an infinity of reals, and an infinity of rationals, and that this infinity was the same quantity, just like in 2 cars and 2 testes, the 2 is the same. However, that argument could also be applied to 2 cars and 3 apples, they're both quantities, thus 2 and 3 are the same. It's a semantic trick based upon your lack of rigour.

Now, for a very long time, mathematics was content with finite cardinals, and the lumping together of everything that wasn't finite as infinite with no more thought. That is perfectly acceptable, and infinite cardinals do not contradict this view point, but in this view we do not qualify what we mean by an infinite quantity as something other than finite.

Cantor, perhaps, noted that two sets have the same (finite) cardinality when and only when there is a bijection between them, and decided to see what happens if we extend this to be a definition for cardinals of infinite sets.

There is absolutely no way that I can make sense, at the level you're arguing from, that "infinity is a quantity like 2" What is "infinity", what is "like"? What, for that matter is "quantity"? Mathematics has rigorous deduction from definitions. Not speculative mutterings without explanation about how everyone else is wrong, whilst simultaneously admitting that you haven't actually read anything that was written about it.

You really ought to pay attention quite closely to Hurkyl's post about how we do not use "infinity" very much. In fact it is perfectly possible to do all of mathematics without actually using the word infinity. It would be more long winded: saying something tends to infinity is a short hand for "grows without bound". The point at infinity, is shorthand for saying the unique point in the one point compactification of C that isn't in C. Adding +/-infinity to the reals is embedding R into a linearly ordered uncountable space with a max and min, and the universal such embedding.

 

[shrumeo]

The problem is that you didnt' bother to explain what you meant by quantity. You said in effect that there is an infinity of reals, and an infinity of rationals, and that this infinity was the same quantity, just like in 2 cars and 2 testes, the 2 is the same. However, that argument could also be applied to 2 cars and 3 apples, they're both quantities, thus 2 and 3 are the same. It's a semantic trick based upon your lack of rigour.

Yes, I see this "argument" as a matter of semantics mostly.

Is 0 a quantity?

0 to me behaves a lot like infinity. In that you can cut infinity in half and you still have infinity. You can double infinity and you still have infinity.

Right, the highest math I took was DiffEq and that was about 10 years ago, so I have no "rigour."

Now, for a very long time, mathematics was content with finite cardinals, and the lumping together of everything that wasn't finite as infinite with no more thought. That is perfectly acceptable, and infinite cardinals do not contradict this view point, but in this view we do not qualify what we mean by an infinite quantity as something other than finite.

Hmm, are you saying that an infinite quantity is finite?
This really is an exercise in semantics.
Maybe we need to come up with some new words.

Cantor, perhaps, noted that two sets have the same (finite) cardinality when and only when there is a bijection between them, and decided to see what happens if we extend this to be a definition for cardinals of infinite sets.

There is absolutely no way that I can make sense, at the level you're arguing from, that "infinity is a quantity like 2" What is "infinity", what is "like"? What, for that matter is "quantity"? Mathematics has rigorous deduction from definitions. Not speculative mutterings without explanation about how everyone else is wrong, whilst simultaneously admitting that you haven't actually read anything that was written about it.

In my head, we can say that I am treating infinity as if I were "counting" to infinity. Or treating zero as if I was "counting" to zero.

You really ought to pay attention quite closely to Hurkyl's post about how we do not use "infinity" very much. In fact it is perfectly possible to do all of mathematics without actually using the word infinity. It would be more long winded: saying something tends to infinity is a short hand for "grows without bound". The point at infinity, is shorthand for saying the unique point in the one point compactification of C that isn't in C. Adding +/-infinity to the reals is embedding R into a linearly ordered uncountable space with a max and min, and the universal such embedding.

Yes, I think there is a problem with wording here.
It sounds like a lot has been made of "infinity" that is much more than plain old "forever."

I'm coming from the grade school perspective here in that I just see infinity as the counting that never ends (like zero is the counting that never starts.)

I don't have the luxury of having math degrees behind me to know the history of the philosophy of number theory and set theory and all that.

I didn't mean to anger anyone by interjecting what I said.
But from my perspective it looked like the question was about how many kinds of zero or infinity or 2 there are. And in my little brain, I just see these things as counting, like on one's fingers.

If the "infinity" this thread adresses is anything but that, please forgive my intrusion into this discussion. I'm only on this Earth to learn (well, mostly).

 

[shrumeo]

The main thing about the various ways to define the word "infinite" is this: when we say X is infinite, we mean that in some sense, the magnitude of X is greater than that of any integer.

It is much rarer to talk about "infinity". Infinite things are rarely called infinity.

----------------------------------------------------------

The main point to remember about the extended reals is that the two points at infinity (+∞ and -∞) are only expected to behave properly with respect to the ordering.

Any other properties they have are a nice bonus. For example, the + operator can be extended "to infinity" by continuity -- specifically, arithmetic has nothing to do with this extension.

You can do the same with the * operator... but one of the places to which * cannot be extended is 0 * ∞.

-----------------------------------------------------------

There are arithmetic methods that permit infinite numbers that behave properly with respect to arithmetic. But, in all of those cases, 0*x = 0, even for infinite x. (Why? Because they behave properly with respect to arithmetic. )

-----------------------------------------------------------

The "infinities" of set theory (that is, the infinite cardinal and ordinal numbers) are, again, simply a type of ordering, which are based on functions between sets. (order preserving functions, for the ordinal numbers) There are ways to define arithmetic of these things by analogy, but it's very strange. For instance, if a and b are infinite cardinals, a + b is simply their maximum. For ordinals, a + b is usually different than b + a.

0 * a = 0 for any cardinal, though.

Believe me when I say that this explains a lot to me, thank you!
I never thought about 0*infinity not making sense, but there it is.

So anyway, thanks for the lesson!

O wait, I just realized that you said 0*infinity = 0.
But infinity * x = infinity ,doesn't it? I would think this leads to a paradox.

Hmm, maybe I need more lessons, but I won't ask for them here.
It will just clutter the real discussion.

Doh! got it. If you have zero infinities you have zero and if you have infinite zeros, that's still zero, so 0*∞=0.

(Yes,but why is 0! = 1 ?) Anyway, just ignore me.....

 

[hypermorphism]

... And in my little brain, I just see these things as counting, like on one's fingers.

If the "infinity" this thread adresses is anything but that, please forgive my intrusion into this discussion. I'm only on this Earth to learn (well, mostly).

The act of counting objects other than your fingers using your fingers is a bijection between the objects you are counting and the counting numbers (represented by your fingers). Sets that can be counted in this fashion are called countable sets, and infinite sets that can be indexed in this fashion are called countably infinite. For example, you can index the set of all even numbers easily: one bijection is twice the counting number. Ie., I can say the first even number is 2, the second even number is 4, and so forth. Conversely, if you name an even number, I can say that that is the nth number in the sequence, using my fingers. Ie., the even number 50 is the 25th number in my bijection/count.
It was a surprise to learn that there were common sets that were not countably infinite, and thus the idea of different kinds of infinity made its way into common mathematics. The proofs involved lie in the section of this thread you claim not to have read.

 

[dextercioby]

(Yes,but why is 0! = 1 ?) Anyway, just ignore me.....

We can't ignore you...Not yet... 0 factorial IS DEFINED AS BEING UNITY...Chosen by convention if you prefer.Because of that,very many expressions in mathematical make sense...Why did they chose 1 instead of any other NONZERO (so we can have factorial of zero in the denominator,as well),i don't really know.It certainly had nothing to do with gamma Euler which gets into shape after u define 0!=1.

Daniel.

 

[matt grime]

0 is the additive identity in a ring, it is nothing like "infinity", whatever you may have decided infinity is (but haven't stated rigorously). Hypermorphism tells you about countable, a little. Essentially, a set is countable if there is a way of picking out one element for every natural number in a way that exhausts the set. Some sets cannot be "counted" like that.

Mathematicians haven't made anything out of infinity that isnt' reasonable, but other people have.

And when hurkyl said 0*infinity=, he is not talking about things you think he is talking about. he is talking abour cardinal arithmetic.

There is also the word 'not' missing from my post, which has lead to the misunderstanding in the second part of your reply using my quoted text. "we do not qualify an infinite quantity as being anything other than NOT a finite quantity." That makes more sense.

 

[danne89]

Hurkyl:
I don't understand your argumentation that 0 * H != 0, H infinity
In the hyperreal numbersystem you've define infinitesimals that multiplied with a infinity large number can become both a finite and a infinitesimal big number.

Inutively you can also see this is correct. Concider a line. It's a infinite extantion in space, but a width of 0. Thus its area is zero.

No paradoxes, which I can dig up.

 

[Hurkyl]

Hurkyl:
I don't understand your argumentation that 0 * H != 0, H infinity
In the hyperreal numbersystem you've define infinitesimals that multiplied with a infinity large number can become both a finite and a infinitesimal big number.

I think you misread me, I said 0 * H = 0 in the hyperreals.

 

[mathwonk]

your persistence has lured me back. i cite the earliest recorded example i know of a bijection: the cyclops in ulysses "counted" off the captive men in ulysses band, as they went outside the cave, by putting over one rock into a pile for each man. as they returned he put the rocks back. thus he knew if all the men had returned even though he apparently did not know how to count! i.e. he knew there were the same number of men as rocks, but did not know what number that was.

this is an example of comparing the sizes of two sets without actually knowing how many elements are in the sets. this also occurs in comparing infinite sets.

 

[arildno]

your persistence has lured me back. i cite the earliest recorded example i know of a bijection: the cyclops in ulysses "counted" off the captive men in ulysses band, as they went outside the cave, by putting over one rock into a pile for each man. as they returned he put the rocks back. thus he knew if all the men had returned even though he apparently did not know how to count! i.e. he knew there were the same number of men as rocks, but did not know what number that was.

this is an example of comparing the sizes of two sets without actually knowing how many elements are in the sets. this also occurs in comparing infinite sets.

Cool example!
I'd like to mention that sheep-herders often used a stick with scorched/scratched marks on for each sheep in his care as a simple counting device (such sticks have been found for example in the Swiss Alps).

I think it is rather interesting that perhaps the most useful counting device in modern maths (bijective correspondence) is, in fact, humanity's oldest counting device..

 

[shrumeo]

So, when we a typical person uses numbers in a typical fashion (let's say in a physics equation) the person draws the numbers from one set (and from what I've seen it's usually the Complex numbers and then even mostly the Real numbers).

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 (?)

So maybe it's not the set of numbers involved, but the "counting device" alluded to throughout?

LOL! the above post doesn't make any sense because .3333~ isn't infinitly large, and neither is .9999~. if they where, then your above post wouldn't hold any credibility because .9999~=.3333~, which by your own argument means that that statement is false (because you said .99999~>.33333~ which means that .9999~=/=.3333~).

Edit: now that i look at your post again, it looks like you are arguing that .9999~ is on a greater level of infinity because .3333~ is smaller than .9999~. what you are arguing is not what the original argument was (and your argument still doesn't make sense). he was arguing that R should have a greater infinity than N (? i think its N, he was talking about the set of all integers) because N doesn't represent as many numbers as R does. the problem is that both R and N are uncountable, which means that neither one can have a greater infinity than the other.

This is what I was talking about earlier.
I'm sorry but in the time I've alotted myself to read over this (not much)
I haven't been able to grasp Cantor's diagonal method, so there must be something deeper to this there.

But above, the thing about 0.3333~ and 0.9999~
These numbers are obviously not infinite. I mean 1/3 is not infinity.
There are an infinite number of 3's behind the decimal and and an infinite amount of 9's. You know they are infinite because you never stop tick tick ticking as you count the numbers (using your fingers).

Just like the original question.

You go counting numbers between 1 and 2 and the numbers between 1 and 1000 and you have the same amount (infinity) of numbers.

If you go from 0 to infinity or if you go from negative infinity to infinity you have the SAME amount of whatever, infinity.

That's why I said I don't see the difference.

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?"

NO. You never stop counting so it's infinite.

Size, you ask? How big are your fingers?

 

[mathwonk]

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.

 

[matt grime]

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 (?)

So maybe it's not the set of numbers involved, but the "counting device" alluded to throughout?

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

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.

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.''

You go counting numbers between 1 and 2 and the numbers between 1 and 1000 and you have the same amount (infinity) of numbers.
If you go from 0 to infinity or if you go from negative infinity to infinity you have the SAME amount of whatever, infinity.

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

That's why I said I don't see the difference.

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

NO. You never stop counting so it's infinite.

Size, you ask? How big are your fingers?

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

 

[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.