Aha! I knew that would get your attention! Well Cantor knew the answer to this question, so I start with a version of his theorem. :

For any set S, the cardinality of S is strictly less than the cardinality of its powerset P(S).
(the powerset P(S) just all possible subsets of S).

Let's use |S| for the cardinality of S. Now, remembering that S and \emptyset are always subsets of S, for some finite |S| you can easily convince yourself that |P(S)| = 2^{|S|}.

The question now is, is this true when |S| = \infty?

A set is said to be countable if its elements can be placed in one-to-one correspondence with the elements of a subset of the "counting" numbers \mathbb{N}. As \mathbb{N} is always a subset of \mathbb{N}, and as \mathbb{N} is infinite, we have the quaint expression "countably infinite" .

We'll assume here that S is countably infinite. So the question is: when |S| = \infty and |P(S)| = 2^{|S|} what is the meaning of \infty = 2^{\infty}?

The one-to-one correspondence I referred to above means that whatever we can prove for \mathbb{N} will by true for any countable set, so we want to prove that |P(\mathbb{N})| \gt |\mathbb{N}|. (Yeah I know, I am using "greater that" rather imprecisely)

Proof: We proceed by contradiction, that is we start by assuming that P(\mathbb{N}) is countable

Let's assume we can make a countable list L of all possible subsets of \mathbb{N} By assuming that L is a countable list, we are asserting we can assign a unique index from \mathbb{N} to each element in the list. Let's say that the n-th element of L is l_n

We form the subset D under the condition that the number n is in D iff n is a member of the n-th element of L, i.e.

n is in D iff n is in l_n.

Now D is a subset of \mathbb{N} and, by our supposition, must be on our list. Let's index this subset according to our rule for forming D, say D = l_p and we have the result that p is in D iff pis in l_p. As this is D, by definition, we have the startling result

p is in D iff p is in D.

How amazing! But let me remind you we are assuming we have written an exhaustive list of the subsets of N, this will become important

Now, for any subset of \mathbb{N} we can find its complement, that is, the subset whose elements are not in the first subset. Let's do this for D and say that C is the subset whose elements are not in D. This is definitely a subset of \mathbb{N} and must therefore, by our supposition, be on our list, so let's give it an index, say l_q.

Note by the definition of D, if C is its complement, q cannot be in D, and must therefore be in C Then by the above

the number q is in C iff q is not in l_q. But this is C! So

q is in C iff q is not in C.

Aaargh!This is clearly bonkers. Where did I go wrong? Well only by making the original assumption that P(\mathbb{N}) is countable, which must therefore be false. So \mathbb{N} is countably infinite, P(\mathbb{N}) is uncountably infinite, which a "bigger" infinity. Cantor's Theorem is proved. Isn't it sweet?

Anyone got any other equally simple and elegant proofs, post them here!

I always thought it was.....

http://www.infiniti.com/ :D

What is infinity?

The more and more that I think about "infinity," the more frustrated I get with the word itself. In my opinion, there is no single thing we would call "infinity." Maybe we can say this is an infinity or that is an infinity (notice the indefinite article "an"). But why, you might ask. Two reasons. The first is that any set NOT being finite is said to be infinite (by definition). Being infinite means that it cannot be counted finitely. The word infinite is an adjective while infinity is a noun... hence they have different definitions.

Second reason. Why not say this set has size equal to infinity? Because if you claim that the size of the set of integers is infinity, I can easily construct a second set that's larger than your first one. It's infinite, too. But we can't say they both have size infinity for this would imply both sets are the same size.

Years ago I thought there was a single concept of infinity. There isn't. In fact, there are an infinite number of infinities. To make things even more confusing, I can create a power set of the set of these infinities. This one is even bigger. But then I can make it even bigger. It goes on and on.



As for the OP wanting us to supply elegant proofs. Hrm. I have read many proofs but I cannot recall one off the top of my head that I was super impressed by. My reactions is usually something like "oh that's neat" or "very clever" but not "OMFG wow...."

Cantor's diagonal proof is incredibly elegant, in my opinion. However, while it showed an important result on its own, I think the more important aspect is the diagonalization proof method itself.

This has played an absolutely invaluable part in especially computability theory and foundational logic. Gödel's incompleteness theorem, Turing's work on undecidability, and almost every limiting result in computability theory uses diagonalization (and simulation) in one form or another.

Infinity is something that never ends, so it can never become infinite because it would require infinite time. Infinity can only exist in the timeless zeropoint that is now, not in reality/time.

524376 is not closer to positive infinity than the number 43. Both are infinitely far away from infinity, that's why infinity can be any number. It's undefined (Joker card).

Proof that infinity is 0: Infinity has no ending and thus no beginning, like zero. Zero is the beginning and the end, alpha and omega. There can't be positive or negative infinity, only neutral...

Proof #2: a circle has infinite polygons, which is the same as saying that it has zero polygons.
Proof #3: 1/infinity=0. 1/0=0.
Proof #4: infinity/infinity=anything (undefined). 0/0=anything (undefined).
Proof #5: you can't add (or subtract) to infinity, just like you can't add to 0. infinity*23467=infinity. 0*23467=0.

Proof #5: you can't add (or subtract) to infinity, just like you can't add to 0. infinity*23467=infinity. 0*23467=0.

-2 + 2 = 0

I'm not sure if I'm catching your direct meaning on this last point.

Cantor's diagonal proof is incredibly elegant, in my opinion. However, while it showed an important result on its own, I think the more important aspect is the diagonalization proof method itself.Sure, I could not agree more - in fact that was the point of this thread, or something like it - an invitation to post proofs that took your breath away.

I have read many proofs but I cannot recall one off the top of my head that I was super impressed by. My reactions is usually something like "oh that's neat" or "very clever" but not "OMFG wow...." Then, my friend, you have no soul!

Then, my friend, you have no soul!

Or maybe because I don't feel the same now that I did in the past about the same proof, I don't recall it. I do remember getting excited over Cantor's proofs MANY years ago. I would sit on the toilet reading them in amazement.

Now, I don't really care because I use the same proof technique all the time.

Proof that infinity is 0Counter proof: 1 +0 = 1\;\;1+\infty \ne 1. Case closed.

Infinity is something that never ends, so it can never become infinite because it would require infinite time. Infinity can only exist in the timeless zeropoint that is now, not in reality/time.

Did you even read my post? Your definition of infinity and infinite are different than what we use in mathematics. Carry your philosophies elsewhere (preferably the Philosophy subforum).

524376 is not closer to positive infinity than the number 43. Both are infinitely far away from infinity, that's why infinity can be any number. It's undefined (Joker card).

Ehh. When you get right down to pure definition of a metric space, you aren't permitted to "compute" d(x, \infty). However, you could take the limit to infinity and get infinity as your answer. But, what if I use the discrete metric space? Then d(x, y) = 1 if x \neq y and d(x, y) = 0 otherwise. Then what? ALL numbers are equality close to the outer reaches of your set..

It all depends on how you want to describe the notion of distance.

Proof that infinity is 0: Infinity has no ending and thus no beginning, like zero. Zero is the beginning and the end, alpha and omega. There can't be positive or negative infinity, only neutral...

Proof #2: a circle has infinite polygons, which is the same as saying that it has zero polygons.
Proof #3: 1/infinity=0. 1/0=0.
Proof #4: infinity/infinity=anything (undefined). 0/0=anything (undefined).
Proof #5: you can't add (or subtract) to infinity, just like you can't add to 0. infinity*23467=infinity. 0*23467=0.

You have no business in the physics and math forum.

Question, if 0 = \infty, then consider the communitive ring \mathbb{Z}_{n} = \mathbb{Z}/n \mathbb{Z}. Clearly, 0 \in \mathbb{Z}_{n}. Where's infinity?

Ok, fine. It's not an infinite set. Ok then, how this:
R = [-1, 1] \subset \mathbb{R}. Addition is defined as the following: \forall a, b \in R, a \oplus b = \frac{a + b}{2}. Multiplication of two elements is defined as the usual multiplication of numbers.

This is a subring of the real numbers (under my created addition and usual multiplication) and hence a ring itself. The set is infinite and 0 \in R. Where's infinity?

Ok fine, R is infinite but it's bounded. Ah ha! You caught me. Well, not really. Why? Let's prove a little "theorem" (notice the quotes temur and others...).

"Theorem 1.1"Let \mathbb{R} be a communitive ring with 1. If 0 = \infty then for all subrings S of \mathbb{R}, \infty \in S.

Proof: By definition of a ring, 0 \in \mathbb{R}. And since all subrings of \mathbb{R} are also rings, 0 \in S. Since 0 = \infty, we have \infty \in S. \box

What's wrong with this? Besides containing some handwaving magic, this "theorem" claims that all subrings of \mathbb{R} are unbounded. However, we just saw an example of a bounded subring of \mathbb{R}.

I doubt you have a response.. but the creation of this post was fun.

Counter proof: 1 +0 = 1\;\;1+\infty \ne 1. Case closed.

Ouch.

Proof #4: infinity/infinity=anything (undefined). 0/0=anything (undefined).

Oh really? How about... 1/1 = 1 and 2/2 = 1. Therefore, 1 = 2?

Counter proof: 1 +0 = 1\;\;1+\infty \ne 1. Case closed.

that doesn't prove anything. i don't know what the = with the / means though but i guess it means "not equal to".

d(x, y) = 1

i only know multiplication, plus, minus, division and geometry. everything else is unnecessary to know (like algebra) because there is no practical use of it (except sounding smart maybe)...d, r and y have nothing to do with math... they are letters.

Your definition of infinity and infinite are different than what we use in mathematics.

ok, but who has the correct definition?

Oh really? How about... 1/1 = 1 and 2/2 = 1. Therefore, 1 = 2?

yes, but 1+1=2 and 2+2=4 so your logic fails. look at the 5 proofs i wrote and you can clearly see the connection between infinity and zero and you will realize that they are the same.

proof #6: imagine a blank paper where nothing (zero) is drawn. what could be drawn there? answer: everything/anything (infinity).

Can infinity exist?

i only know multiplication, plus, minus, division and geometry. everything else is unnecessary to know (like algebra) because there is no practical use of it (except sounding smart maybe)

If you don't know mathematics beyond that which my 13 year old sister can do, how can you claim to have the knowledge and insight to tackle problems thousands of years old (nature of zero and infinity).

And there are practical uses of mathematics beyond your adolescent knowledge. Group theory is used in cryptography. Graph theory is used to analyze networks and brain activity. Calculus/analysis are used in physics.

ok, but who has the correct definition?

This question could easily take an unsuspecting victim. Firstly, definitions in themselves are not right or wrong. If they were, it would imply that definitions have a truth value and can be used to justify statements. This brings me to my second point. Definitions exist merely to make communication of ideas more efficient. If I had to explain what a train looked like every time I asked a friend whether he wanted to drive his car or take the train, then it would take a very long time. Definitions also allow us to standardize things. All mathematicians know what a metric space is, what a derivative is, what a rational number is, etc.


yes, but 1+1=2 and 2+2=4 so your logic fails. look at the 5 proofs i wrote and you can clearly see the connection between infinity and zero and you will realize that they are the same.

You're logic fails. How does 1/infinity=0. 1/0=0. imply 0 = infinity? In fact, how does 1/0 = 0 at all? If you are trying to prove 0 = infinity and you claim that 1/infinity = 0, you can't substitute 0 in for infinity to say 1/0 = 0. At this point, I think you try to claim 1/infinity = 1/0 implies 0 = infinity.

THAT is what you call faulty logic. You can't assume to be true what it is you are trying to prove.

proof #6: imagine a blank paper where nothing (zero) is drawn. what could be drawn there? answer: everything (infinity).

This isn't even a mathematical proof.

Explain to me how you can draw, to scale, the same sheet of paper WITH a man standing to the left all on this blank sheet of paper you originally started out with?

That can't be drawn. Sorry.

i only know multiplication, plus, minus, division and geometry. everything else is unnecessary to know (like algebra) because there is no practical use of it (except sounding smart maybe)

Do you really believe this? Consider that the computer you are using couldn't possibly have been built using only the 4 basic arithmetic operators and geometry...

May I ask how much geometry you actually know?

...d, r and y have nothing to do with math... they are letters.

Letters are ubiquitously used as symbols in math. Sometimes we use letters which have a relation to the concepts we define (such as d for distance), sometimes they can be pretty arbitrary and merely convention based (such as variables usually being called something like x or z).

Can infinity exist?

good question, you are not far away from understanding that infinity is zero (non-existent).

If you don't know mathematics beyond that which my 13 year old sister can do, how can you claim to have the knowledge and insight to tackle problems thousands of years old (nature of zero and infinity).

because you don't need to know much math to understand them, just logic... besides, those problems have been solved thousands of years ago by indians.

definitions in themselves are not right or wrong. If they were, it would imply that definitions have a truth value and can be used to justify statements.

this definition is wrong: "5 is a number 2 times bigger than 3"

How does 1/infinity=0. 1/0=0. imply 0 = infinity?

since you get the same result with infinity and zero it implies that they are the same. you don't get the same results with two other numbers, like 5 and 7, which means that they are different.

if you make a polygon with infinite sides or zero sides you get the same result: an undefined shape (like a circle).

In fact, how does 1/0 = 0 at all?

it's basic division: if you divide 1 cake into 3 you get 3 parts, and if you divide it into 2, you get 2, and 1 if you divide it into 1... so logically, it must divide into 0 (ie. infinite parts) if you divide it with 0.

May I ask how much geometry you actually know?

i know how to calculate area, volume and circumference of polygons. i mite know some more but i don't remember now.

In a countably infinite set, all the ordinals (index of a list) are finite, aren't they?

Does this mean a set with infinite cardinality, like the integers, can't contain any infinite ordinals (as identifiers of their place in the list), since there is always an ordinal larger than the largest of a countable set?

P.S. How about... 1/1 = 1 and 1/2 = 0.5; Therefore, 1 = 0.5?

because you don't need to know much math to understand them, just logic... besides, those problems have been solved thousands of years ago by indians.



this definition is wrong: "5 is a number 2 times bigger than 3"



since you get the same result with infinity and zero it implies that they are the same. you don't get the same results with two other numbers, like 5 and 7, which means that they are different.

if you make a polygon with infinite sides or zero sides you get the same result: an undefined shape (like a circle).



it's basic division: if you divide 1 cake into 3 you get 3 parts, and if you divide it into 2, you get 2, and 1 if you divide it into 1... so logically, it must divide into 0 (ie. infinite parts) if you divide it with 0.

Oh ok. I didn't know.

Where can I learn more? What books do I buy? What classes can I take? Who teaches this?

This definition isn't wrong:

"5 is a number almost 2 times bigger than 3"

(he he)
logically, it must divide into 0 (ie. infinite parts) if you divide it with 0.What must divide into zero? You can't divide zero by anything (at all)--because you can't find a part of absolutely nothing (nada, ning, zilch, zip, zero).

And you can't divide anything "with" zero, either. You can't find a non-existent part of something (matter, mass, etc).

This definition isn't wrong:

"5 is a number almost 2 times bigger than 3"

(he he)

It isn't even worth it man :(

lol.

Infinity is a concept beyond our understanding

What must divide into zero?

a cake which you divide into 0 pieceS...

You can't divide zero by anything (at all)--because you can't find a part of absolutely nothing (nada, ning, zilch, zip, zero).

nothing is just the state where everything is in balance, so if you divide it, it will become unbalanced and become anything.

And you can't divide anything "with" zero, either. You can't find a non-existent part of something (matter, mass, etc).

scientists say that all matter is 99.x% (100%) space, nothing, so if you divide it infinite times (zero) you get nothing.

Can infinity exist?

If you don't know mathematics beyond that which my 13 year old sister can do, how can you claim to have the knowledge and insight to tackle problems thousands of years old (nature of zero and infinity).

And there are practical uses of mathematics beyond your adolescent knowledge. Group theory is used in cryptography. Graph theory is used to analyze networks and brain activity. Calculus/analysis are used in physics.

Explain to me how you can draw, to scale, the same sheet of paper WITH a man standing to the left all on this blank sheet of paper you originally started out with?

That can't be drawn. Sorry.

Well, give me some time, as I am still counting. (That's my "adolescent" proof).

Learned

Let's see: "a cake which you divide into 0 pieceS...", would have to be zero bits of cake.

"nothing is just the state where everything is in balance", my non-existent bit of cake is also balanced. What about two kids on a see-saw?

"all matter is 99.x% (100%) space, nothing, so if you divide it infinite times (zero) you get nothing." Infinite times, is not zero. As for matter being mostly empty space, how do you divide empty space? We can't see atoms to divide them, anyway

Can infinity exist?

Take electric circuit

V= I x R

where V= voltage, I= Current, R=Resistance

Whenever there is applied voltage but there is no current (zero) , it means resistance is infinite.

This is one way to imagine presence of infinity.

P.J.LAKHAPATE
plakhapate@rediffmail.com

OK, the title of this thread was a bit of a tease; let's try and return to (my version of) sanity.

One says that a measure has been defined on a point set S iff one can associate to it some non-negative real number, that is m(S) \ge 0, such that, where A,\;B \subset S

m(A \cup B) = m(A) + m(B) - m(A \cap B). In other words,

m([a,b)\cup (b,c] =m([a,c]).

So let S=(0,1), an uncountable set, by Cantor's diagonal argument. And now suppose that A\subset S denotes the irrational elements in S, also uncountable by the same argument. Then the rationals may be denoted as B = (0,1) \setminus A. Note that B is countable.

Then, since there is some irrational that approaches arbitrarily close to 0 and to 1, then m(S) = m(A) so, in m(S) = m(A \cup B)=m(A) +m(B), we must conclude that m(B) makes no contribution to the sum, and therefore that the rationals is a set of measure zero.

This is true of any countable set.

(Hmm, this is rather sloppy reasoning, but the conclusion is at least correct)

By "irrational", you mean a number that can't be expressed as a ratio or quotient of integers (or natural numbers)?

Do any of the irrationals in (0,1) have an ordinality? S must have infinite (uncountable) ordinality, like the reals?

How is (0,1) defined, or is it just two arbitrary points in a larger set?

By "irrational", you mean a number that can't be expressed as a ratio or quotient of integers (or natural numbers)?Yes, precisely that.
How is (0,1) defined, or is it just two arbitrary points in a larger set?Uh, (0,1) is a subset of R (with the usual ordering), that is, it's an interval that contains every possible real which is strictly greater than 0 and less than 1 - i.e. every real, rational and irrational, between 0 and 1 but excluding exactly 0 and exactly 1. Cantor showed us this is an uncountable set. P.S. by edit: on brief reflection, I don't think you need to specify any particular ordering on (0,1) - any ordering will suffice, I guess

As it happens, you don't need to use the interval (0,1) for this subset; (1,2), or (53,100) would do just as well for this argument to apply.

Then, since there is some irrational that approaches arbitrarily close to 0 and to 1, then m(S) = m(A) so, in m(S) = m(A \cup B)=m(A) +m(B), we must conclude that m(B) makes no contribution to the sum, and therefore that the rationals is a set of measure zero.
(Hmm, this is rather sloppy reasoning, but the conclusion is at least correct)
Very sloppy reasoning. Just as there is some irrational that approaches arbitrarily close to zero, so is there some rational. The rationals, like the irrationals, are dense in the reals. This particular argument just doesn't cut it.

Divide the interval (0,epsilon) where epsilon is some non-positive real into a countably infinite number of disjoint subintervals such that every point in (0,epsilon) is in one of the subintervals. The overall interval has measure epsilon. Because the partitioning covers every point and because the partitioning is countably infinite, the measure of the partitioned set is also epsilon.

Because the rationals in (0,1) are a countable set, this set of intervals can be mapped to the rationals. The measure of the rationals in (0,1) is bounded from above by the sum of the measures of these intervals, and that measure is epsilon. As epsilon can be made arbitrarily small, the rationals in (0,1) (or any subset of the reals) form a set of measure zero.

The rationals in [0,1] have Lebesgue measure zero but are not Jordan measurable.

Although I am not sure what this has to do with our discussion of infinity, which was very informative to me. :)

What about the irrationals in (0,1)? They don't have measure zero?
The rationals don't contribute to the measure of a set but the irrationals do? And rationals are a countably infinite set? Measure just means "sum", or "total number"?

P.S. Any ordering means any series (like a series of inverse primes, say), or combination (superposition), in fact there is an infinite possible number of orderings? But any series, however defined, has ordinality, and no term has infinite ordinality, or in a countable or uncountable set with infinite cardinality, no member has infinite ordinality - there is always a last term in any set...?

Think of measure as an extension of the concept of length. The measure of any simple open or closed interval in the reals is exactly equal to the length of the interval. The measure of a finite number of disjoint intervals is the sum of the lengths of the intervals. The same goes for a countably infinite number of intervals.

Lebesgue measure is additive. Since the rationals in (0,1) have measure zero and the interval itself has measure one, the Lebesgue measure of the set of irrationals in (0,1) is also one.

Frud, please be more precise in your use of mathematical terms. A mathematical series is a single number. A single number does not have "ordinality". Ordinality is a property of a set and comparison operators on that set. You're PS is a bit too jumbled to make sense out of it.

Isn't it simpler to say infinity in math is the point at which a particular model or equation breaks down? It's an error more than an answer, a way of saying "That equation can't be used in this situation".

Like plakhapate's example with resistance. It doesn't mean there's infinite resistance in the circuit, it means you can't use that equation to measure current because the circuit is not closed. It's impossible for electricity to flow through an open circuit, so it's impossible to calculate the current. In other words, resistance is undefined, not infinite.

Look at it mathematically, rearrange V = I x R for R. R=V/I. If I is 0 and R is infinite we have infinity=V/0. But division by zero is not infinity, it's undefined, therefor resistance is undefined, because you can't use this equation to model an open circuit.

Infinity is simply an unbounded, uncountable set. It really doesn't need to be any more complicated than that.

good question, you are not far away from understanding that infinity is zero (non-existent).



because you don't need to know much math to understand them, just logic... besides, those problems have been solved thousands of years ago by indians.



this definition is wrong: "5 is a number 2 times bigger than 3"



since you get the same result with infinity and zero it implies that they are the same. you don't get the same results with two other numbers, like 5 and 7, which means that they are different.

if you make a polygon with infinite sides or zero sides you get the same result: an undefined shape (like a circle).



it's basic division: if you divide 1 cake into 3 you get 3 parts, and if you divide it into 2, you get 2, and 1 if you divide it into 1... so logically, it must divide into 0 (ie. infinite parts) if you divide it with 0.



i know how to calculate area, volume and circumference of polygons. i mite know some more but i don't remember now.

Yorda... Zero accounts to nothing no matter what you strain it under. For instance, take this:

(n^2)^0...

In algebra, you work out the products so it makes:

n^2.0

and that makes to no odds for the power, so we are left with the one value:

n^0 = 1

See how zero played no part for us?

Stop conflating the terms infinite (an adjective) and infinity (a noun).

The term infinite has one meaning in mathematics, not finite, but there are multiple ways a set can be infinite. I suggest googling "Hilbert Hotel" as a very good place to start.

The term infinity has multiple meanings in mathematics. It can mean an unbounded limit in calculus, the point/line/plane at infinity in topology, or the cardinality of some infinite set.

Very sloppy reasoning. Just as there is some irrational that approaches arbitrarily close to zero, so is there some rational. The rationals, like the irrationals, are dense in the reals. This particular argument just doesn't cut it.You're quite right, I don't know what I was (un)thinking of.
What about the irrationals in (0,1)? They don't have measure zero?
The rationals don't contribute to the measure of a set but the irrationals do? And rationals are a countably infinite set? Measure just means "sum", or "total number"?If I give a slightly more extended version of the proof D H gave, you should see the answer to your question. As I know that mathematics is not your strong suit (not an insult, btw, merely an observation), I will try and talk you through it.

First note that the subject of this thread, if there is one, is related to the established fact that the rational real numbers are infinite in number, but countable, whereas the irrational reals are uncountably infinite. This looks weird at first sight, right?

The technical version goes like this: a set is said the countable if it can be placed in a one-to-one correspondence with a subset of the set of "counting" numbers 1, 2, 3, ...... One calls this set \mathbb{N}. (Note that some people include zero in this set. I won't in what follows). Perhaps counter-intuitively, the technical definition of a subset requires that any set is a subset of itself!

By R one usually means the elements of the set \mathbb{R} arranged in a "line" in the obvious way (note the different fonts). A segment of this line may or may not contain its endpoints. Where it does one writes [a,b] \in R, a closed interval, where it doesn't one says (a,b) \in R, an open interval. Note that, if x \in (a,b) one may say that (a,b) is a subset of [a,b].

So. Let (a,b) be an open subset of R. Now let \{x_i\} denote the rationals in this set. Since the rationals are countable, I may offer each x an index from \mathbb{N}, where x_n denotes the n- th rational in (a, b).

I will now "contain" each x_n in an open interval of measure \frac{\epsilon}{2^n}, where \epsilon \gt 0. You should see why I said that order is not important; all we require is that the n's match

Now recall that the union of two sets is itself a set, which by the axioms, may contain each "member" only once; that is, the union of [1,2] and [1,3] is [1,3]. Write [1,2] \cup [1,3]=[1.3]. One says that the two sets on the LHS are non-disjoint. Notice this crucial fact; the measure of [1,2] is 1, the measure of [1,3] is 2, so the measure of the union of non-disjoint sets is less that the sum of their individual measures.

So. Since epsilon is not zero, and since the real line is continuous ("no gaps"), we may assume that the open intervals containing the x_n are not all disjoint. Therefore the measure of the union of these intervals must be less that the sum of their individual measures.

A quick exercise with pencil and paper should easily reveal that the sum of these measures, when n is taken to be 1, 2, 3, ......, is \epsilon. So we have the following: \sum^{\infty}_{n=1}\frac{\epsilon}{2^n} = \epsilon, therefore the measure of the union of all open intervals containing our set \{x_i\} of rationals is less that epsilon. But all we require of epsilon is that it be non-zero, that is, it can be as small as we like and then smaller still, so we must conclude that the measure of the rationals is zero, as this is the only number we can can think of that is smaller that any other number.

Finally note that this only works if our set can be indexed by a counting number, so the irrationals are not a set of measure zero. Sorry to be so long-winded.

Stop conflating the terms infinite (an adjective) and infinity (a noun).

Didn't I bring this point up already? :p

Didn't I bring this point up already? :pLooking back, I see that you did, but I missed it in the midst of all the psychoceramics on the first page. Maybe the wackos will pay attention to me ... (although I probably have a better chance of winning the lottery--without even buying a ticket)

With respect to this business of dividing by zero. This is just due to inefficiencies of the English langauge and dread semantics. Unfortunately words cannot express the true nature of this problem.

Barry

re: Division by 0: What about Nullity? :)

http://en.wikipedia.org/wiki/James_Anderson_(computer_scientist)

(I'm kidding)

http://upload.wikimedia.org/math/0/e/6/0e62a24a2b9d43be8dbc85cb531b9063.png

A mathematical series is a single number.
A mathematical series is a set of terms, an ordered set, with a first, or zeroth element, and a list of successor elements (if you understand the concept of a list as a data structure, then it's defined recursively).
A series can resolve to a single discrete value, like zero, or to something that isn't a discrete value (like infinity). It isn't accurate to describe a series as a single number. A series certainly has ordinality, it's ordered.

A mathematical series is a set of terms, an ordered set, with a first, or zeroth element, and a list of successor elements (if you understand the concept of a list as a data structure, then it's defined recursively).
Get your terms straight! That you cannot do so, and that you argue when corrected, is one of the many reasons some have noted that you are weak on math.

A mathematical series is not a set of terms. You are thinking of a mathematical sequence, which is indeed an ordered set of terms. A mathematical series is the the sum of all of the elements in the sequence. The sum, if it exists, is a single real number, not a set. Numbers by themselves are not sets, and thus have no cardinality. (And stop using ordinality when you mean cardinality! Ordinality is not a word.)

Numbers by themselves are not sets, and thus have no cardinality. (And stop using ordinality when you mean cardinality! Ordinality is not a word.)Cardinality is the measure of how big a set of things is. Ordinality is the place in the set (if it's an ordered set).
A list is an ordered set, so is a series or sequence (actually all three are equivalent in meaning, to most people).

Ordinality, means: "what place" something has in a list, series, or sequence (of terms). You could line up a number of objects, and start at one end of the line, call the end object the "first" object, then the next object is the "second" object; the first two objects have an ordinality (place number) assigned to them which indicates their "part" of a list, and this indication (ordinality) can be extended to the remainder of objects in the same list. A list already implies an ordering, or sequence, just by being a list.

This is what I mean when I use the word "ordinality". Maybe it isn't in your dictionary, but it's a word, and it means something.
Your semantic distinction between list, series and sequence is trivial, and kind of meaningless (except in advanced math?).

In the infinite sequence or series of integers, starting say at 1, and ignoring 0 or negatives, so the ordinality (position in the sequence) corresponds to the integer value, how many are infinite (or have an infinite ordinality)?

A list is an ordered set, so is a series or sequence (actually all three are equivalent in meaning, to most people).
This is the physics and mathematics forum. Physicists and mathematicians use precise language because they have learned that failure to do so leads to miscommunication and misunderstanding. Mathematicians use two different words because sequences and series are different things.

An example of a sequence: \{9/10,9/100,9/1000,\cdots\}. A series: 9/10+9/100+9/1000+\cdots = \sum_{r=1}^{\infty} 9/10^r = 0.999\cdots = 1. Series and sequences are related in the sense that one can generate a sequence from the partial sums of a series. The partial sums of the above series form the sequence \{9/10, 99/100, 999/1000, \cdots\}. Both of the sequences I described are examples of Cauchy sequences of the rationals.

Your semantic distinction between list, series and sequence is trivial, and kind of meaningless (except in advanced math?).
Understanding the distinction between "sequence" and "series" is essential to basic freshman calculus. Freshman calculus is not advanced mathematics by any means.

In the infinite sequence or series of integers, starting say at 1, and ignoring 0 or negatives, so the ordinality (position in the sequence) corresponds to the integer value, how many are infinite (or have an infinite ordinality)?
None. The sequence does not magically end at infinity. Like the Energizer Rabbit, the sequence keeps on going and going. "Infinity" is not a member of the sequence.

Brilliant. I took calc 101, but I don't recall that the word "sequence" was used much at all. Instead, "series" and "sums" (also limits) were what we discussed. A list is something you tend to see in IT courses.
Your example of a sequence looks a lot like a list. Another term I remember was: "tuple".

P.S. of course there are no infinite members in an infinite set, except if the set happens to be a set of infinite series, some of which diverge...?

P.P.S. Infinity does not "magically" end a sequence? An infinite sequence has no "end"; I don't remember being told about any "magic"...

A mathematical series is not a set of terms. You are thinking of a mathematical sequence, which is indeed an ordered set of terms.

A series is a sequence. Convergent series converge to the limit of the partial sums, which are considered to be a sequence.

Doh. Stupid math teminology..

I take that back. A series is not a sequence. I looked it up and a series is an ordered pair of sequences.

I take that back. A series is not a sequence.
Good so far ...

I looked it up and a series is an ordered pair of sequences.
Smart ass. Being obtuse is not helping these people who have some very basic misunderstandings. Moreover, this definition wrong. Two arbitrary sequences do not form a "series". There is a grain of truth in this incorrect definition: One can use two sequences to represent a series. A series is the sum of all of the terms in a sequence. The partial sums of these terms (e.g., the first term, the sum of the first two terms, the first three terms, etc) is another sequence. This sequence of partial sums in a sense also represents the series.

Smart ass. Being obtuse is not helping these people who have some very basic misunderstandings.

?

Moreover, this definition wrong.

I needed Latex to convey what I meant accurately but I did not think it mattered that much since anyone can look up the definition. That's why I omitted the details. If I wanted to be obtuse I wouldn't have admitted I had gotten it wrong in the first place, now would I? I'm not sure why you're attacking me here but you're only exposing your own misconceptions. I didn't specify what kind of sequences make up the ordered pair but that doesn't change the fact that a series is (or can be defined) as a certain ordered pair of sequences. Therefore there was nothing wrong about the description I gave. Perhaps it wasn't the whole picture but it certainly wasn't wrong. No less wrong than me saying a linear space is an ordered triple of a set and two functions.


If you wanted clarification you could have said so and I would have given it. I don't see how this involves being a "smart ass" or being "obtuse".
Leave that language to the dolts here who think they know everything and be more constructive.

Infinity is undefined. To be defined means to be (de)finite. Infinity can never be reached, infinity can never be... something... so all it can do is expand forever.

1/0.001=1000
1/0.0001=10000
1/0.00001=100000

The closer we come to zero, the closer we come to infinity, but since infinity has no beginning nor ending, it can't be positive or negative, it can only be nothing, 0.

Well, I must say, I have to side with SouthStar here.

Given any set S, a sequence in S may be defined as a map \mathbb{N} \to S, where \mathbb{N} has the usual ordering, such that, for any m \le n \in \mathbb{N},\; s_m \le s_n \in S.

Suppose now there is an binary operation, say, for example +, defined on S.
Under this circumstance, there is a map \mathbb{N} \to + such that +_m \le +_n, that is adding three times is greater than adding twice, say. This is a sequence, as defined above.

So now suppose that s +_m = s_i \le s+_n = s_j \in S. This supposition makes no sense, unless I know what I mean I mean by s_i \le s_j
Therefore I need to support this supposition by defining another map \mathbb{N} \to S, again with the usual ordering.

In other words, a series may be considered to be a pair of sequences.

The pair of sequences SouthStar wrote about are the sequence of terms and the sequence of partial sums. So yes, a sequence can be analyzed in terms of two sequences: two sequences related by means of the operator '+' on S. My point is that it is invalid to merely say a series is a pair of ordered sequences. Two arbitrary sequences in some set S do not necessarily form a series.