math question

[dimava]

what is x if:

x * x = x
x + x = x
x - x = x

note: there are at least 2 possible answers

x = 0

[Fox]

x = x :)

or x = -0


Dont't see another solution...

[dimava]

Originally posted by dimava
note: there are at least 2 possible answers

[dimava]

correct Dennis and Fox althogh there are more possible answers

[Guv]

Zero was fairly obvious. Infinity works also. Negative zero does not work.

Every computer I ever heard of which allowed negative zero, made it become positive zero when you did arithmetic. Also, the product of two negatives is always positive.

[dimava]

correct GUV

[Fox]

inf - inf = 0...

[Sam Finch]

I'd like to disagree with Guv (and I'd assume dimava) and state that infinity doesn't work.

when we try to eveluate an expression with infinity in it we don't just substtute infinity into it, we have to work out the limit of the function F(x) = x - x as x tends to infinity.

and as x - x = 0 we see that F(x) = 0 and so the limit of F(x) as x tends to infinity also = 0, not infinity.

This isn't actually the most rigorous way of working it out, but it does produce the correct answer.


one more rigorour was of showing that x - infinity does not work is by defining y = 1/x (which if x = infinity = 0

so 1/y - 1/y = 1/y

(1-1) / y = 1/y

multiply both sides by y (this only works if y is not infinity)

0 = 1 if y is a real number

and hence y cannot be a real number and so x must be 0.

[dimava]

ok, I guess I mis read the posts, the 2 and only correct answers are 0 and -0

[Sam Finch]

- 0 is also wrong, as Guv pointed out

-0 * -0 = +0.

*sits there and looks smug, then realises that rather than proving himself to be the greatest mathematician of all time he's just proved that he'd a pedantic Bastard, but continues to look smug anyway*

*If Einstein or anyone else walks into a bar I won't be happy*

[Guv]

Sam, a bit nit-picky for this context. Perhaps the question did not require the answer to be correct for all possible functions approaching infinity. Now I am attempting to prove that I am right by picking nits.

[Michael]

If we want to get picky then here's my contribution.

I don't agree that -0 is a different answer to +0. They are the same.

The original post is titled "math question" (I won't get picky about that quaint American habit of omitting the "s" on "maths"!!!) In mathematics there is no difference between -0 and +0.

Anyone who has a computer that treats these as 2 different values should go and get their money back.

[Guv]

Sam, are you sure you do not have a typo or two in your post? On my second scan, it seemed a little strange. I never read it carefully: I view this as a tongue in cheek post, not to bbe taken seriously.

Michael, once upon a time there was a computer which used signed magnitude numbers internally and could represent zero with a plus or a minus sign. I think it was a National Cash Register Co.(NCR) computer. it was not so terrible as you might think.

For numeric operations, there was no difference between plus & minus zero (excellent design concept). However, minus zero did not pass a machine language test for zero (not a very good design concept).

There were a few department stores which used NCR systems due to using NCR cash registers, and being familiar with the company. This was not a good criteria for using NCR computers (they were kludges).

Some of my friends and I amused ourselves when we discovered that due to some sloppy programming, the credit card billing program could be fooled into dunning a customer for an overdue balance of zero dollars & zero cents. It could be done by creating a credit balance, and then charging an item costing the same amount (EG: Charge a $50.00 item when you had a $50.00 credit balance). The billing program made credit balances negative and then added the charges. It turns out that 50 added to minus 50 resulted in minus zero. The program that printed your bill would print minus zero as $0.00, but the subroutine which which decided what to print checked for a credit balance code (print amount as a credit), checked for a zero balance (send no bill), else send bill for an amount due. If you had no activity on your account you would start getting nasty dunning letters demanding a zero payment. They would go away when actually turned over to a collection agency.

I have seen letters in the Ann Landers column claiming that you had to send a check for zero dollars and zero cents to clean up your account. These were embellishments on an otherwise true story. Actually, almost any activity would stop the dunning letters, and sooner or later a human being would notice if you stopped using the account.

[Sam Finch]

Guv

I think one or two typos is a bit of an understatement (in fact saying that one or two typos is a bit of an understatement is a huge understatement on my part so I'll forgive you for understating)

did you get x-x = x to work for x is infinite btw? I'm pretty sure you won't. :)

[Michael]

Guv: I think basically we are in agreement.

I have not heard this story about NCR. It always amuses me when a major company such as NCR makes fundamental errors such as this in their software.

Oh well, it just proves the point that so long as there are programmers around, there will always be programming errors.

That reminds me about a cock-up I made at work today but I won't embarass myself in public by telling you guys about it!!! At least I found it before it went out!!!

[Guv]

Sam, without trying to be too mathematically rigorous, I suspect that subtracting the infinity associated with all the even integers from the infinity associated with all the integers would result in the same transfinite number (I think this is Cantor's word for infinity).

My knowledge of Cantor's work is sketchy at best, but I think the following is correct, but not mathematically rigourous.

I have seen proofs that the set of all even integers can be put into one to one correspondence with the set of all integers, as well as the set of all odd integers.

This anomaly might be the definition of a transfinite set. If not the definition it is surely a property of such sets.

I also think I remember that sets which can be put into one to one correspondence have the same cardinality. Hence, All the integers, the even integers, and the odd integers are assigned the same cardinal number (Is this term correct?). Id est: They are the same infinity.

Hence, I think I have X-X = X, where X is Aleph-Null (the first transfinite number, associated with all the integers).

Stuff like the above bugged me no end when I first encountered it, especially since I was mathematically very naive at the time.

If you disagree, I will not argue further. However, I do not think you should disagree without reviewing a book which includes some of Cantor's work on transfinite numbers.

[Guv]

Sam, I never felt confident in my knowledge of transfinite numbers & related theory. I will trust your last post, but still have nagging doubts.

Somewhere it seems to me that I read about infinity times infinity, infinity plus infinity & infinity minus infinity all being infinity, with infinity divided by infinity being either ambiguous or requiring the use of limits to define.

I suppose infinity minuus infinity was not included in whatever I read.

If at some time in the future I discover that you are wrong, I intend to look for a hitman who will waste you for $5.00.

If you ever discover that are wrong and think there are cheap assasins available, start being paranoid!

[Sam Finch]

I think infinity - infinity is also ambiguous, but in this case it's 0 (because we used limits to find it)

Infinity's a bit of a wierd subject, I thought I had it until I found out that 1 to the power of infinity is e. Which means that 1^(2*infinity) = e^2 and 2*infinity does not equal infinity.

I'm pretty sure I'm right on this one, I tried a couple of other methods as well and they all came out as 0.

If it does turn out I'm wrong then I'll add you to the list of people who are trying to kill me.

[Guv]

Sam, I would like to see a citation on 1 to the power of infinity being e.

I do not think I could refute or verify that on my own, unless perhaps if threatened with death or promised a large sum of money. At least I would not touch this problem with a 3.048 meter pole, without strong motivation.

Without an expert opinion, I find it hard to believe. For the time being, I will be agnostic on this one.

Do not get me wrong. Unlike my opinion of certain individuals who shall rename nameless here, I respect your opinions and knowledge of mathematics.

[Sam Finch]

I have to say that I can't remember exactly how to work it out, when I say I Found out I meant I read it in a big maths book, unfortunatley it's one of those books that are so huge that if you don't take down a page number you'll never see that page again. But it was in a proper maths book.


it uses the idea that


Lim F(x) Lim F'(x)
x->y -------- = x->y ----------
G(x) G'(x)



and works it out like that, unfortunatley I can't find an F(x) and a G(x) that work for it, I might try to find that page again for you. (there are also rules on what F and G can be which I can't remember)as I said my immediate reaction was that it didn't make sense on account of 2*Infinity not being equal to infinity anymore. But it was in a maths book and that's not the sort of thing they'd print if there was a chance it was wrong.

[Guv]

Sam, if 2*Infinity is not infinity, what the hell is it? Some finite number? Aleph-1? The power of the continuum? Your aunt Suzie's beagle? An imaginary number? Perhaps a quaternion.

I missed the above anomaly in your earlier post. There is something seriously wrong here, Sam.

[kedaman]

Einstein walks into the bar and picks up his laptop with vb installed on it.

Now if you open vb and in immediate window write

?null*null

what do you get?

Also try:
?null-null
?null+null
and why not
?null/null

And -0 is 0. because 0*(-1)=0

I think infinity can be compare with null, if you substract a infinity from infinity then you get infinty :) But eh,

Lim X -> oo x-x = 0
doesn't mean that
X=oo x-x=0

Also
Lim X-> 0 1/x = oo
but
Lim X-> oo 0/x is not 1

[Lafor]

saw this too late but..


Fact: Infinity is not a number...

So a sttement like 1 + infinity or something
similar is meaningless in a mathematics

Some number can "go toward infinity" or as the french put it "...tend vers l'infini"

It was not really a point for discussion...
It is a fundamental fact ...

[Guv]

Laffor, perhaps the term "infinity" in all possible contexts cannot be considered a number. However, I think there are "transfinite numbers" which are the cardinal numbers associated with various infinite sets. For example, "Aleph-0" (or perhaps "Aleph Null") is the cardinal number for the of the set of all integers. "Aleph-1" is the number associated with the set of all subsets of the Aleph-0 set, and so on. The "power of the continuum" is the number associated with the set of all real numbers. I think that Aleph-1 = "power of the continuum" is an unproved conjecture that everybody believes is true. I once thought I had a simple proof for this conjecture, but am sure there is something wrong with it (it was so simple that, if valid, some one would have thought of it before I was born).

I am not certain of all the above statements, but are fairly sure that they are close to being valid.

Over 100 years ago, a man named Cantor wrote a book or some papers about infinite sets or transfinite numbers or something like that. I think he is credited with being the first to formally deal with this subject matter. It is not an area I feel comfortable with because it was one part of a one semester course I took in prehistoric times.

It seems to me that he defined "cardinal numbers" Aleph-0, Aleph-1, et cetera associated with various infinite sets. I think he first defined an infinite set as a set which could be put into a one to one correspondence with a proper subset of itself. He established that two sets had the "same number" of members if the members could be put into one to one correspondence with each other.

Aleph-0 (or Aleph Null) was the smallest "transfinite number" and corresponded to the set containing all the integers. He proved that the same transfinite number was associated with the set of all even integers and with the set of all rational numbers. He proved a lot of other counterintuitive theorems not applicable to finite sets and finite numbers.

While "infinity + 1" might not be valid statement (in the absence of a lot of disclaimers & defintions), I think that "Aleph-0 + 1 = Aleph-0" is a valid statements

I do not think it is correct to claim that infinite numbers (Id est: Transfinite numbers) are not numbers, unless you mean to say that "Infinite numbers are not finite numbers." At any rate there is a body of mathematical knowledge which deals with infinite numbers in a manner analogous to finite numbers.

[Sam Finch]

Lafor

Technicly you are right, but as with all maths there's a way around it. Where we write infinity what we mean is
F(c) where F(x) -> infinity as x -> c.

so for an equation containing infinity we substitute F(x) in to the equation to get G(x) and find the limit of G(x) as x -> c. However whatever we do must be shown to work for all F.


And as we can say F(x) - F(x) = 0 for all F,x we can say that for x = infinity x - x = 0.

For Aleph - 0, Aleph - 1 etc we can't realy use them as numbers without being very careful indeed, with a lot of proof at every stage.

I have to say I also thought I had a proof that |R| = Aleph - 1, involving solution sets to infinite numbers of simultanious equations, but apparrently it cannot be proved (something called the Continuum Hypothosis, which seems stange that nobody will look at a proof because there is a hypothosis that one does not exist)


Just as a point to note, I didn't find the page in the book, but I did find this.


The First 2 parts of the Taylor Series for exp(h) are

exp(h) = 1 + h

and hence

lim(x->0) exp(h) = lim(h->0) 1 + h
lim(x->0) h = lim(h->0) ln(1 + h)
1 = lim(h->0) ln(1 + h) * (1/h)

e = lim(h->0) (1 + h) ^ (1/h)


you can try this for different values of h as h->0 (use a calculator rather than VB) and the smaller value you use for h the closer you get to e, for h = 0 you get 1^infinity.

what I meant by 2* infinity does not equal infinity I meant that we had a function F(x) where F(infinity) is a real number where F(2*x) = F(x)^2. and F(infinity)^2 does not equal F(infinity). Which seems a bit strange.

[Guv]

Sam, this thread has pulled me into places where I do not feel comfortable. Now that I am here, my obsessive compulsive nature keeps me here trying to make sense out of the various posts.

First, let me repeat what I said in a previous post: Mathematics relating to Cantor's work is just not an area in which I trust my knowledge. Having said that, I want to further state that I am very suspicious of your knowledge in this area.

I just do not trust either of us. That having been said, let me try to bring up some issues.

Yes, it seems that X-X = zero for all X, which suggests that infinity - infinity = zero. However, I am not sure that it is valid to just plug infinity into some equation without paying any attention to the many infinities available. It seems to me that infinity - infinity is likely to be as indeterminate as zero/zero and infinity/infinity.

For example, consider the infinity associated with all the real numbers. This is provably larger than the infinity associated with all the integers. If subtracting the latter from the former is a valid operation, the result is clearly not zero.

Now having warmed up with some non rigorous remarks, let me at least try to be rigorous. What about the following
1)N = Cardinal number of members in the set of all integers from 1 to N, inclusive. This is essentially a definition of Cardinal number.
2)Define E = Cardinal number of even integers in same set.
3)Define X = Cardinal number of odd integers in same set.
4)Now, for all N the following seem to be true. N = E + X, X = N - E, and E = N - X

When N is allowed to grow without bound, Cantor's Aleph0 (aleph Null) is the cardinal number associated with all three of the above sets, At least it can be proven that the same cardinal number is associated with the infinite version of all three of the above sets.

Hence Aleph0 = Aleph0 - Aleph0

I think I can develop a similar proof that Aleph0 - Aleph0 = zero.

As I said, perhaps (infinity - infinity) is indeterminate/ambiguous, just like 0/0, infinity/infinity.

I disagree with proof that "X - X = X" can only be valid if X is zero. I think that it is valid only if you assume that X is finite.

You posted the following.
one more rigorour was of showing that x - infinity does not work is by defining y = 1/x (which if x = infinity = 0

so 1/y - 1/y = 1/y
(1-1) / y = 1/y

multiply both sides by y (this only works if y is not infinity)

0 = 1 if y is a real number

and hence y cannot be a real number and so x must be 0.
First, you are merely stating "X - X = X" in an unnecessarily complicated fashion. The proof you are looking for should go as follows. Consider the equation "X - X = X"
1)Zero obviously satisfies the equation.
2)If X is any finite value other than zero, you can divide both sides by X.
3)This gives 0 = 1, a contradiction.
4)Hence, X cannot be any finite value other than zero.

Note that if X = infinity, X/X is ambiguous/indeterminate, so division by X is not allowed in this case. What you have proved is that zero is the only finite value which satisfies "X - X = X"
Further analysis is required for infinite values.

I am not certain of the above, only somewhat comfortable.

Next issue for discussion: I do not believe that 1^infinity = e, and I feel very sure about it. You posted the following.
lim(x->0) exp(h) = lim(h->0) 1 + h
lim(x->0) h = lim(h->0) ln(1 + h)
1 = lim(h->0) ln(1 + h) * (1/h)

e = lim(h->0) (1 + h) ^ (1/h)
I agree with the final statement, but deny that it can be used to claim that 1^infinity = e.
First, the above is unnecessarily confusing. The exponential function (e^x) is defined as or proved to be the limit of [1 + x/n)^n] as n approaches infinity. This leads to e= limit[(1 + 1/n)^n]. Using h = 1/n and the rest of the above is merely confusing.

The general proof involves the expansion of (1 + 1/n)^n: The kth term is C(n,k)*1^(n-k)*(1/n)^k Where C(n,k) is the combination of n items taken k at a time. Note that each term can be shown to be a fraction with denominator containing n^k*k! (k factorial). The numerator contains a power of one times a polynomial of order k. Determining the limiting series involves the following logic

1)The power of one in each term equals one.
2)The first term of the polynomial is n^k. when divided by n^k, this becomes one.
3)In the limit, the n^k in the denominator causes all but the first term of the polynomial to vanish.
4)Hence, the general term is 1/k!, resulting in the following.
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! . . . . .

Note that assigning e= 1^n for n = infinity, would make e the first term of the above series. It would also make the second term e, and the third term would be e/2.

[Guv]

Sam, I forgot about the following. In reply to something I posted, you said.
I have to say I also thought I had a proof that |R| = Aleph - 1, involving solution sets to infinite numbers of simultanious equations, but apparrently it cannot be proved (something called the Continuum Hypothosis, which seems stange that nobody will look at a proof because there is a hypothosis that one does not exist)
I believe that proofs have been attempted, and serious mathematicians are quite willing to look at such proofs if the proposer has some mathematical credentials.

I think the continuum hypothesis is the statement that Aleph1 equals the power of the continuum. I think some mathematician asserted that it was true. I believe that various mathematicians tried to prove it. I think they all failed, but most believe it to be true. I remember the above from prehistoric times. My memory is generally pretty good, but I would not bet large sums of money on it.

For those not familiar with some of the above terms.
1)There is a set consisting of all the integers. Aleph0 (by definition) is the cardinal number (or infinity) associated with that set.
2)There is a set of all subsets of the set of all integers. Aleph1 is the infinity associated with this set (obviously a bit larger than Aleph0).
3)There is a set containing all the real numbers. It can be proven to have more members than the set of all integers. The infinity associated with this set is "The power of the Continuum" (a definition).

I think it can be proven that the power of the continuum is not Aleph0, but must be Aleph1, or Aleph2, or Aleph3, et cetera.

[Lafor]

Infinity is not a number.....

think for one second about this...

If infinity is a number then what is its value...?

What is the biggest number there is?

For every x, there exists a x+1 > 0

I could go into the whole of ALEPH0, but we do not need to
go there at all to establish this

infinity, as we've learned from the basics of math (and it is well-established fact), is not a number. Often represented by the "tired 8" that is lying down.....

If time permits, I will proceed later on

[Lafor]

Thanks

[Lafor]

In lay-man: terms "L'infini n'est pas un nombre!!"

Infinity is (from the start) a mathematical abstract notion
"Infinity" as referenced is the "property" of the set of integers pertaining to this fundamental fact:

For every given number there is another one which is
greater than that number

This characteristic proves that there CANNOT exist a number that is greater than all numbers

In fact, if we were to find such an object "infinity", then
it would not be a number since it would not admit a number greater than it

As a result, to say that a certain variable that it is
infinity is really an abuse of language that leads to serious confusion

Ex:
One knows you cannot divide by zero, one would be incorrect to state that the result is "infinity"

The correct proposition is that in a division, the smaller
the number by which we're trying to divide, the greater the result of the division

Whoever says "NUMBER" excludes "infinity" which is Not a number.... The real world cannot be put into contact with the theoretical "infinity"

If the "infinity notion" has a certain utility in mathematics
it has NONE in physics... As a matter of fact, the apparition of "infinity" in physics (i.e the attempt to put a number on top of it) signals a flaw

At the beginning of this century, electrical theory was incapable to explain the existence of Hydrogen atom, the simplest chemical element... According tho the then-formulas, the force of attraction would augment when the charges approach each other....Then (to make a long story short), came "infinity" in this context... That was enough to have the theory dismissed...

It had to be replaced by.... a new one "quantum mechanics" (mechanique quantique) ....

Vale!

[Guv]

Lafor, I consider you to be correct in saying that infinity should play no role in physics. At least I hope it never does. Perhaps there are others that will disagree. However, it seems to be necessary to deal with it in the world of mathematics. Furthermore, I think that physics make good use of some mathematical concepts derived from the use of infinity.

The theories dealing with black holes and Big Bang cosmology talk about singularities. I suspect that it will be discovered that quantum physics and relativity theory break down short of the singular conditions they predict for the start of the universe and the center of a black hole. A mild caveat: Physics is getting ever closer to "ultimate truths" and/or the limits of human understanding. So maybe the singularities will remain.

By the way, I always wondered how all the matter in the universe could get out of a space smaller than an atomic nucleus, while absolutely nothing gets out of a black hole formed by ten or more solar masses. Black holes do evaporate, but the Big Band could hardly be considered to be evaporation. I have read talk about the Big Bang being due to space expanding rather than matter expanding into existing empty space. I have read about concepts like that, but I understand them as well as I understand how a woman's mind works.

Back to mathematical infinity. First, I do not believe that Cantor claimed that there is some greatest integer. After n, there is always an n+1, and I do not believe that Cantor claimed otherwise. He did claim that "the set of all integers" is a thinkable concept, and set out to think about it. Prior to Cantor, mathematicians had been using concepts relating to infinity in an intuitive, non-rigorous fashion. Cantor formalized thoughts about infinity, or at least made them less sloppy. He not only assigned a number (Aleph0) to the number of members of the set of all integers. he further showed that there were larger numbers, like the number of real numbers.

Infinity seems necessary for many proofs, especially those dealing with series approximations. If terms like (n^2 + 5n + 4)/n^2 could not be treated as equal to one when n grows without bound, there are a lot of handy theorems which could not be proven. Some of those theorems are used in physics. If quantities like 1/n are treated as zero in the limiting case, you are implying that n is infinite.

In a sense, infinity is no less real that numbers like the square of two (In the world of physics, you cannot have such a number, only an approximation). You cannot even have two in the world of physics, only an approximation to it. Worse yet, what about the square root of minus two? This certainly does not seem to exist in the world of physics. Yet imaginary arithmetic is quite useful in vector analysis, and physicists are happy to work with vectors. The roots of most polynomials beyond the 5th order have roots which can be approximated, but never can be expressed as numbers. In some sense, their existence is an abstraction. Such numbers merely seem more knowable than infinity.

Without Cantor's transfinite numbers, how do answer questions like how many integers are there? How many rational numbers? How many real numbers? Are there more real numbers than rational numbers? Are these unallowed questions like "what is a 4-sided triangle?" or "How high is up?"? Do you merely say those questions have no answers. The question comparing the number of reals to the number of rationals certainly seems to have an answer. If it has an answer, can you deny the existence of infinite numbers?

When you say that infinity is not a number, I disagree. If you say it is not an integer or not an "ordinary number" (What ever that is), tres bien! It is not a real number, nor an algebraic number, nor a transcendental number, nor a lot of other kinds of numbers.. It is a lay term for a transfinite number, which mathematicians find to be quite useful at times. I object to denying the existence of such numbers as used by mathematicians.

[Lafor]

Guv....

Cantor reasoned this way:

The cardinality of a finite set is the snumber of elements
in the set. So, for infinite sets, he reasoned that we should extend the definition. And he continued: The cardinality of a finite set is an integer, that of an infinite set is ALEPH (hebraic notation). And this is how he designated the transfinite numbers

3 = 2 + 1 therfore is the cardinal of {@, {@@}, {@,{@@}}}, und so weiter...

The initiator of Cantor's set theory was really DEDEKIND

The problem with Cantor's theorem:
IF (E) stands for the set of parts of set P(E), so

Card E < Card P(E)

As a result, Cantor himself noted (1899) an anomaly:
The set of all sets cannot be a set
Because if such a set X were to exist, then (X) would be
element of X and therefore

Card P(X) < Card X

but per Cantor's own theorem

Card P(X) > Card X

which would be a CLEAR contradiction

So, the question:

"Is The set of all sets a set?"

Let's attempt to answer this question.....
Be careful here... with the given

"In a small town there is ONE and ONLY ONE barber.
The barber shaves all the people who DO NOT SHAVE THEMSELVES
Q. Does the barber shave himself?"

In the same vein...
Can we cook up the catalog of all catalogs? The resulting
"thing", is it a catalog?

We can stop here... Whe have to...

Cantor did not imply or say in his works that infinity
was a number. He was trying to deal with the concept and
did well but not in the vein of our discussion... Cantor
-- transfinite numbers!!

Everything is a matter of definition (in Math)
How do we define INFINITY? And, based on the admitted, definition, it is NOT a number....

A gentleman, named Plato, once said: (I am attempting a translation here -- forgive me)
"The ?number? 'Infinity', alone, is susceptible of all kinds of divisions!"

I think we can put this to rest for now....
But as for the BIGBANG thing... we'll point out the holes
later on

Vale....

[Guv]

Lafor, Cantor used the concept of "The set of all subsets of a given set," which does not lead to contradictions. He was not guilty of the "Set of all sets" error. Dedekind (of "cut" fame) did some original work on set theory, including some notions relating to infinity & infinitesimals, but I believe that Cantor is considered the first to formalize notions of "transfinite numbers."

The Barber of Seville definition is an example of a subtle circular definition, only if stated properly. Something like The Barber of Seville is that citizen of Seville who shaves all citizen who do not shave themselves and only shaves those who do not shave themselves. Without the bolded phrase, the barber can be a trained monkey or a citizen of a neighboring town, or some other non-citizen, thus dodging the paradox. formal logic does not allow circular definitions in order to avoid such problems. In fact, formal logic requires some undefined terms.

I understand your thoughts about "infinity" as used in many contexts not being a number. To me, there is a problem with denying that "transfinite numbers" are not numbers. Consider one concept mentioned in a previous post of mine.

I believe that the following statement is true and meaningful: "There are more real numbers than integers." In the abstract world of mathematics, it is considered provable. If that statement is allowable for consideration, it seems intuitively true. The statement seems to imply the ability to quantify the members of the set of all integer and the set of all reals. This implies more than one type of infinite set, and implies that an infinite set has something very like a number associated with the concept of how many members are in the set.

[Lafor]

Just to clarify...
Ok... I think we're branching off a bit...
1. I pointed out that Cantor himself (1899) discovered the
anomaly..(Barber..) usually referred to as Russel's paradox...and (Cantor admitted it and wrote about it as did Russell in response to Frege)....

2. The reference to Dedekind [I made] (in the stated context) [besides being true] is an admission of mathematical history

3. Transfinite numbers, we can leave alone in this context

Make it a good day..

[robbie73]

I would just like to say that: ALT+F4 = end to infinte loop.

And before anybody commments about necromancy, the website started it when I clicked search!

[Foxer]

Masterful thread necro.


Unlike my opinion of certain individuals who shall rename nameless here, I respect your opinions and knowledge of mathematics.

I see some people were onto Woss as far back as circa 2000.

[wossname]

April 99 in fact.

[MaximilianMayrhofer]

To say that X-X=X is ambiguous for X = infinity is ridiculous. Infinity is not to mathematics as quantum is to newtonian. It still follows the same rules. Infinity is a concept, used to represent a very large real number. Any real number subtracted from itself is always 0, regardless of its size.

Even if we accept that there are different levels of infinity, such as the summation of all even integers vs the summation of all odd integers, the fact is that the question states very explicitly 'x-x', which can be translated into "regardless of what form of infinity we have represented with x, we are obviously using the exact same form of infinity during subtraction because otherwise we could not have used the same variable 'x' ".

[robbie73]

just to throw a spanner in..time is infinite. yet cyclic.

time minus time =???

lol

[BillGeek]

Why does everyone bring up such stupid questions anyway? Every single time I open one of these darn threads, my eyes start to ache, my brain ceases to function, and I generally sit in front of the computer with a blank stare... like now...