Can infinities have different sizes?

[DoubleMike]

I've had a discussion with two friends about infinity. I am of the opinion that infinities come in different sizes and some infinities can be bigger than others. My argument goes somewhat like this:

Between the numbers 1 and 2 there are infinite real numbers. Between 1 and infinity there are infinite integers.

However, since for any two successive integers there is an infinite number of real numbers, there are more real numbers than integers.

Despite my best efforts, neither agree with me. They say that infinity is infinite and since "it goes on forever nothing can be greater than it." But I think this definition is somewhat prosaic and has limited mathematical value.

What do you think? Can infinities be of different sizes?

 

[dextercioby]

The German mathematician Georg Cantor wrote something about the theory of infinites.

The set of reals is not COUNTABLE...

Daniel.

 

[DeadWolfe]

Generally, this discussion is more philosophical than mathematical.

In the realm of acceptable math, infinity is not really a meaningful term.

 

[Hurkyl]

What do you think? Can infinities be of different sizes?

It depends entirely on what you mean by "infinity" and "size".

 

[arildno]

Welcome to PF, DoubleMike!
Yes, there does exist different degrees of infinity, but not really in the sense you think!

The basic problem with infinities is that the distinction more/less doesn't make a good distinguishing tool.

Let us look upon the issue from a different angle:
Suppose you've got two sets of objects, A and B, and that there are finitely many objects in A and B.
We want to evaluate the relative "sizes" of these sets in the following manner:
Pair together one object from A with an object from B, and remove both from their respective sets.
Proceed in like manner.

Since you've got finitely many objects, you'll end up at last with one of the following 3 situations:
1. No objects left in A, objects left in B (we say that A had "fewer" objects to start with than B had)
2. Neither have any objects left (we say that A and B started out with equally many objects)
3. Objects left in A, none in B (we say that A had "more" objects to start with than B had)

As you can see, this cumbersome counting technique captures exaxtly what we mean in the finite case of what the words "fewer/equal/more" is supposed to mean.

This pairing-off technique is what we need to use when dealing with sets containig INFINITELY many objects!
Something very surprising happens:
Suppose A is the set of ALL natural numbers "n".
Let B consist only of the even integers.
At first, we would say there were "more" objects in A than B, but see what the pairing technique gives us.
For a given integer "n" in A, we pair that off with the EVEN integer "2n" in B.
EVERY "n" in A are thus paired to an even integer in B, and every even integer in B is paired off with some integer in A!

Is there fewer or more elements in A or B?
As you can see, for the infinite sets, that question doesn't have much meaning; what we can say is:
1) B is a subset of A (any even integer is certainly an integer, so it is contained in A somewhere)
2) There exist a way to pair off A-and B-elements (it is possible to construct a bijection)

To get back to your original idea:
It can be shown that it is impossible to construct a bijection between the set of natural numbers and the set of real numbers.

For natural reasons, a set which can be paired off with the set of natural numbers is called "countably infinite"; the set of real numbers is said to be "uncountably infinte".

So there you have it; there exist different degrees of infinities, but it is rather tricky to understand it..

 

[damoclark]

Yes, that is a very interesting question. I think there are different kinds of infinities. Let me add something if I may.

Consider what happens if you choose two numbers between 1 and 2 and multiply them together, then your answer will always be greater than the two numbers you chose to begin with.

eg 1.1*1.2 = 1.32

1.32 is bigger than 1.1 and 1.2

If you multiplied three numbers together the same principle would apply

eg. 1.1*1.2*1.3 = 1.716

If you multiplied ten number together

eg 1.1*1.2*1.3*1.4*1.5*1.6*1.7*1.8*1.9*2.0 = 67.04425728

the same thing happens.


Now imagine if we could multiply all the numbers betwwen 1 and 2 together. Since there are an infinite number of numbers between 1 and 2, when you multipy all the numbers between 1 and 2 surely your answer should be infinity.

Lets call this infinity(1..2)

Now what would your answer be if you multiplied all the numbers between 1 and let say e = 2.71828..., then surely the answer would be infinity also.
Lets call this infinity(1..e).

Now which infinity is bigger? infinity(1..2) or infinity(1..e) ?

 

[dextercioby]

How would you figure it??What would your answer be and why?Keep in mind that neither R nor any subset of it are countable...

Daniel.

 

[matt grime]

How would you figure it??What would your answer be and why?Keep in mind that neither R nor any subset of it are countable...

Daniel.

N is a subset of R, as is Q, as are (uncountably) many countable sets.

 

[matt grime]

Now which infinity is bigger? infinity(1..2) or infinity(1..e) ?

How are we supposed to know? You've just completely redefined multiplication to allow for infinite operands so we probably shouldn't guess what on Earth you're doing.

 

[dextercioby]

Sorry,i meant intervals... Why the heck didn't i think about N & Q?


Daniel.

 

[matt grime]

The interval [x,x] is a countable* subset of R that is also an interval.


* some people differentiate take countable to mean infinite. some, like me, do not.

 

[eNathan]

I've had a discussion with two friends about infinity. I am of the opinion that infinities come in different sizes and some infinities can be bigger than others. My argument goes somewhat like this:

Between the numbers 1 and 2 there are infinite real numbers. Between 1 and infinity there are infinite integers.

However, since for any two successive integers there is an infinite number of real numbers, there are more real numbers than integers.

Despite my best efforts, neither agree with me. They say that infinity is infinite and since "it goes on forever nothing can be greater than it." But I think this definition is somewhat prosaic and has limited mathematical value.

What do you think? Can infinities be of different sizes?

Mathamaticly, yes, your theory is correct. .999~ > .333~

 

[Gecko]

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.

 

[hypermorphism]

the problem is that both R and N are uncountable, which means that neither one can have a greater infinity than the other.

N is countable by definition. R is uncountable by Cantor's diagonal argument. People usually think of R as "larger" than N simply because they don't have the same cardinality, and N is a subset of R.

 

[DoubleMike]

And what is Cantor's diagonal argument?

 

[hypermorphism]

And what is Cantor's diagonal argument?

The proper form of Cantor's argument without reference to representation can be found at http://mathworld.wolfram.com/CantorDiagonalMethod.html . If this leaves you confused, there is the decimal form found in layman's and introductory texts that relies on your associating real numbers with their infinite decimal representation. This simplified argument can be found at http://planetmath.org/encyclopedia/CantorsDiagonalArgument.html .

 

[cepheid]

The interval [x,x] is a countable* subset of R that is also an interval.


* some people differentiate take countable to mean infinite. some, like me, do not.

Hi,

Just trying to improve my understanding. So if I'm getting what you are saying, a countable set is simply one that can be put in a one-one correspondence with the set of postive integers. When I saw this definition, I thought to myself: "ok, so it means you can *count* the elements in the set basically, you can say this is the first one, this is the second one...etc." So you can assign a postivie integer to each member of the set. There is a one-one mapping. I realize that nothing in there specifies that you have to use ALL of the integers. You can do this to a finite set as well. So in short, a countable set can be either finite or infinite, with at most as many elements as there are positive integers? I was just trying to express what you were saying in my own words. I hope I haven't used any terminology incorrectly.

Thanks.

 

[hypermorphism]

Hi,

Just trying to improve my understanding. So if I'm getting what you are saying, a countable set is simply one that can be put in a one-one correspondence with the set of postive integers. When I saw this definition, I thought to myself: "ok, so it means you can *count* the elements in the set basically, you can say this is the first one, this is the second one...etc." So you can assign a postivie integer to each member of the set. There is a one-one mapping. I realize that nothing in there specifies that you have to use ALL of the integers. You can do this to a finite set as well. So in short, a countable set can be either finite or infinite, with at most as many elements as there are positive integers? I was just trying to express what you were saying in my own words. I hope I haven't used any terminology incorrectly.

Thanks.

Yep. Countable is a generalization of how people count. The terminology "uncountable" just refers to the cases where you can't do what you described in order to cover the entire set.

 

[cepheid]

Ok, so isn't the existence of "countable" vs. "uncountable" sets (different cardinalities) at the heart of what the original poster was trying to say? There are infinitely-many integers, but there are more real numbers than integers...so is that a different type of infinity? Again, maybe the terminology is poor (different infinities), but the idea is sound, right? Our prof mentioned it in passing in class.

 

[hypermorphism]

Ok, so isn't the existence of "countable" vs. "uncountable" sets (different cardinalities) at the heart of what the original poster was trying to say? There are infinitely-many integers, but there are more real numbers than integers...so is that a different type of infinity? Again, maybe the terminology is poor (different infinities), but the idea is sound, right? Our prof mentioned it in passing in class.

I would put "more" in parentheses, but yes. The cardinal number for the naturals is called aleph_null while the cardinal for the reals is bet, or the continuum. There are other concepts of infinity as well: http://mathworld.wolfram.com/CardinalNumber.html .

 

[Galileo]

In the sense of cardinal numbers, there are different kinds of infinity.
In general, if A is a set, then the powerset of A: \rho(A) (which is the set containing all subsets of A) is 'bigger' than A in the sense that there exists no surjection f:A \to \rho(A).

The above can be quite easily and elegantly proved, but for some reason it doesn't seem to work when you take A to be the set of all sets...


Another kind of infinity is 'the point at infinity' in the complex plane. You can add this point to \mathbb{C} resulting in the extended complex plane. This is a totally different kind of infinity ofcourse.

 

[Hurkyl]

for some reason it doesn't seem to work when you take A to be the set of all sets...

Yet more proof that there does not exist a set of all sets!

 

[Galileo]

Yet more proof that there does not exist a set of all sets!

Hurkyl! Just the man I need.

Is the nonexistance of the set of all sets related to Bertrand Russell's paradox? (When A is the set of all sets which do not contain themselves).
How would you show it?

 

[matt grime]

Russel's paradox states that, if we define sets with unrestricted rules, then there is a problem

Namely define X:= { Y | Y is a set and Y is not an element of Y}

Then X is a member of itself and not a memeber of itself, contradiction.

Cantor shows that given a set A there is no bijection with P(A), thus if A were the set of all sets, then, as P(A) is a set, containing A, it must actually equal A, and thus there would be a bijection if there were indeed a set of all sets.

Exercise. Attempt to actually think of a set that contains itself. If you do so, you'll quickly realize how unimportant this actually is.

You should be careful about saying "existence" in such things. We tend to say that given a model of a set theory, the class of all objects that are sets in that theory is not a set in that theory, though it may be a set in some larger universe - that is some other model of the set theory. A set theory is a collection of rules, a model of a set theory is a class of objects that satisfy those rules. Nothing a priori comes with the absolute label "set" round its neck.

This idea may be familiar from other axiomatic systems. For instance, what is a vector (please don't say something with length and direction, we aren't talking physics here)? It is an element of a vector space. What is a vector space? It's a class of objects satisfying certain rules. Is the vector space a vector inside itself? No. Same with set theory.

 

[saltydog]

Density

Infinity and the Real Number System: Very interesting. May I make a proposal?

Why does Mathematics fit nature so well?

Because the Real Number System is dense!

Works for me,
Salty

 

[matt grime]

The reals are dense in what? That makes no sense at all. Denseness is a property of a subset with respect to an ambient space.

 

[mathwonk]

the original work by georg cantor is still appealing on the topic of different sizes of infinity, though about 100 years old by now:

Contributions to the Founding of the Theory of Transfinite Numbers
Cantor, Georg
Price: US$ 4.45 [Convert Currency]

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Bookseller: Harvest Book Company (Fort Washington, PA, U.S.A.)



here is another even more valuable book, covering many more topics:

SET THEORY
Hausdorff, Felix
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Book Description: New York: Chelsea Publishing, 1962. Hard Cover. Very Good/No Jacket. 8vo - over 7¾" - 9¾" tall. Ex-library. Some wear on cover, mostly on corners and spine. Bookseller Inventory #001841

Bookseller: Ivan A Luka (Brentwood, MD, U.S.A.)

 

[robert Ihnot]

Now imagine if we could multiply all the numbers betwwen 1 and 2 together. Since there are an infinite number of numbers between 1 and 2, when you multipy all the numbers between 1 and 2 surely your answer should be infinity.

Double Mike:Lets call this infinity(1..2)

Now what would your answer be if you multiplied all the numbers between 1 and let say e = 2.71828..., then surely the answer would be infinity also.
Lets call this infinity(1..e).

Cantor had the concept of cardinality, relating one set to another. THERE ARE AS MANY NUMBERS BETWEEN 0 AND 1 AS ON THE WHOLE LINE! This can be illustrated by a simple diagram,use your imagination.

(0,1) (1/2,1/2)

(0,0) (1/2,0)

We draw two parallel lines at X=0 and X=1/2 of height 1. Then we construct a line to connect (0,1) and (1/2,0) (If you can visualize that.) This gives us the triangle (0,0), (0,1), (1/2,0). Now consider triangle hypothesis moving along the X axis from 1/2 to infinity.

Then at any point in the advance, the hypothesis cuts the line between (1/2,0),(1/2,1) AT EXACTLY ONE POINT, yet the hypothesis sweeps across the whole positive X axis starting at 1/2. (Well, O.K. move the triangle back 1/2 to get rid of that problem). SO THERE ARE AS MANY POINTS BETWEEN 0 AND INFINITY AS BETWEEN 0 AND 1, so there is no difference in your calculations!

The diagonal argument deals with the difference between the rational and the irrational.

 

[mathwonk]

here is one of cantors diagonal arguments as i recall it from high school, about 45 years ago:

assume there is the same number of real numbers between 0 and 1, even those having only zeroes and 1's in their decimal expansion, as there are positive integers.

that means we canj write down a list (infinitely long) containing all those real numbers.

so we write down first the decimal corresponding to the integer 1, then under it we write down the real decimal corresponding to 2, etc...

this yields a doubly infinite array of the digits 0 and 1. i.e. we have an infinitely long list of decimals, and in each row we have an infinitely long decimal, possibly ending in all zeroes after a while.


Then we will construct another decimal that is actually not in the list as follows:

Just run along the "diagonal" of the list. i.e. the "diagonal decimal" is the decimal whose first entry is the first entry of the first decimal in the list. its second entry is the second entry of the second decimal in the list. its third entry is the third entry of the third decimal in the list...etc...


now with this decimal in hand, go down it entry by entry and change every single entry to the opposite choice. i.e. for each entry which is a 1, change it to a zero, and vice versa.


the result will be a decimal which does not equal any of the decimals in the list, since to differ from a decimkal in the list it suffices to differ in only one entry, and this new decikmal differs from the nth decimal in the list by its nth entry.

now since every list of decimals gives rise to a decimal not in the list, it follows that no such list can contain them all. hence it is not true that there are the same number of such decimals as positive integers.




to abstract this argument, consider the set of positive integers as a given set N, and think about all subsets S of that set. To describe one subset S of N, means for each element of N, i.e. for each positive integer, we must decide whether or not it belongs to our subset S. If it does assign a 1, if not assign a 0. this gives us an infinite sequence of 0's and 1's, lo and behold, precisely an infinite decimal containing only 0's and 1's.



hence if there were the same number of subsets of N as elements of N< then we could write down a list of those subsets and hence of those decimals, each numbered by a positive integer.

we are back where we were before. since there is no such list of decimkals there is no such list of subsets, and so in fact there are mroe subsets of N than there are elements of N.


more abstractly, if N is any set at all, and we find a map f from N to the set of all subsets of N, we can construct a subset S of N, using the map f, as follows: let x belong to S if and only if x does not belong to f(x).

i.e. the "diagonal subset" contains x if and only if x does belong to f(x), and we change it to the opposite subset, where for every x in S, x does not belong to f(x).


then this new subset S is not f of any x, since if S = f(x), then x does not belong to f(x), i.e. x does not belong to S. But by definition of S, if x does not belong to S, then x did belong to f(x). Since we are assuming S = f(x), this is a contradiction.

this argument proves there is no surjection from N to the set of all subsets of N, for any set N at all. hence any set, even an infinite one, has more subsets than elements.

 

[shrumeo]

Hmm, I didn't even bother to read all the posts but it seems to simple even to a non-mathematician like myself.

Infinity is a quantity like "2."
There is only one "infinity" like there is only one "2."
There aren't different kinds of 2 because 2 is just a quantity.

There are infinity Real numbers between 1 and 2.
There are infinity integers between 1 and forever.

An analogy could be made to say:
There are 2 cars in the garage.
There are 2 testicles in my scrotum.

Are there 2 types of "2"?
Nope.

Infinity is just a quantity (albeit an uncountable one).

 

[matt grime]

Well, thankfully you're wrong. But don't let that stop you arguing from a point of ignorance.

 

[hypermorphism]

Hmm, I didn't even bother to read all the posts but it seems to simple even to a non-mathematician like myself.

Infinity is a quantity like "2."

This statement is false. Infinity does not behave like any real number.

 

[mathwonk]

that's it. I'm outta here. (after reading shrumeos post.)

 

[shrumeo]

ok geniuses, explain it to me

why isn't infinity a quantity and 2 is?

 

[Hurkyl]

2 is a real number (or rational number, integer, et cetera) by definition. And, by definition, there is no real number (or rational number, integer, et cetera) called infinity.

 

[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 (+&infin; and -&infin;) 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 * &infin;.

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

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.