Don't ask me. But I know it is of utmost importance in mathematics.

Related links:

Graham's number - Wikipedia, the free encyclopedia

If you follow the links in the following:

Theory of Immensity - The Stripy Strudel's Journal - by Alexey Feldgendler

article, you soon end up in Russia.

P.S.
Here

Prototype JavaScript framework: Prototype 1.6.0.2: Bug fixes, performance improvements, and security

is how I found the last article. It is about Prototype 1.6.0.2: Bug fixes, performance improvements, and security .

Prototype is a JavaScript Framework that aims to ease development of dynamic web applications.

Dan Webb is a London-based web developer is one guru user. Among other things, he has written sumo a gemeric microformats parser for JavaScript.

There is always a side note.

This thread is aimed at being a philosophical and number theoretic thread.

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can it be measured???
how can you say the size of infinite if it cannot be measured???

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Interesting that you answered. You get one positive rep point for that. I did not expect an answer.

The concept of cardinality

"In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {1, 2, 3} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers".

and cardinal number is important.

"In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. For finite sets the cardinality is given by a natural number, being simply the number of elements in the set. There are also transfinite cardinal numbers to describe the sizes of infinite sets. On one hand, a proper subset A of an infinite set S may have the same cardinality as S. On the other hand, perhaps also counterintuitively, not all infinite sets have the same cardinality. There is a formal characterization that explains how some infinite sets have cardinalities that are strictly smaller than other infinite sets".

The cardinal numbers are:

For example the simple Cantor Set is huge.

"The key observation here is that we can specify a mapping, or correspondence, from onto [0,1]. Since is a subset of [0,1], this means that [0,1] and must be the same "size" (or have the same cardinality) -- but let's save the actual details for another day.
Is this cool or what? A set so small that we couldn't possibly draw it on paper (or, in this case, on a computer screen) is still, somehow, as big as an entire interval.
Here's that "big versus small" deal again: The Cantor set is apparently "big", since it has the same cardinality as the interval [0,1] (an uncountable set). If you stick around to read yet more on the Cantor set, you would see that the Cantor set is pretty darn small, too. Of course, if you're tiring of magic and wonder, you could always wimp out".

This is pure mathematics: Very interesting, but my life is too short to study this topic in any detail.

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I do not think infinite actually exists. Infinite is merely a tool to say, 'We do not know'. In math it commonly takes the place for "I do not Know" and there are very many mathematical variations on infinity. 'I do not know' is not as precise as 'infinite' (not that 'infinite' is precise)

One Scientist will tell you today that the universe is Infinite. then the following breath tell you it gets bigger every day.
Infinite - Not a finite thing. If the universe is infinite today' then yesterday it must have at least been somewhat less than infinite. From some point searching for infinity becomes as a religion, some do not believe in it, some do. If you are inclined to a religious belief the acceptance of infinity seems totally acceptable.

I think, size than we could not possibly imagine is a probably a good way to describe it. Or simply look at its varied use in maths. But does it exist. . I personally would rather be dead than live in an infinite place.

The Question is confusing. 'How large is infinite?' It is not big, or small. it is not a finite thing.

Infinity has no boundary or limitations - it is merely a concept that we use as a tool we vary the concept to suit the job at hand. . . infinity has infinite variety, not even bordered by our own imagination..

If you were to define a size for infinity. I could merely imagine it to be bigger and you would simply be wrong... such is the nature of a concept...

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I'm with Tubby. I don't believe infinity can be measured, as it is boundless and that means without boundaries that can be measured ...

So, I don't perceive 'infinity' as a mathematical concept, since it can't be quantified ... nor does it seem to me to be simply what we don't know. Rather I see it as a metaphysical concept that allows us to label the sheer, unutterable limitlessness of all that is (and isn't?). There is no end in either direction. You can neither reach the end nor hit the wall.

It is a difficult concept to comprehend, since our minds *seem* to have limits, but it is an inherently freeing concept, because once one gets even the smallest notion of what infinity means, one is led to understand that there are always an infinite number of possibilities, and that makes it possible for one to create whatever one desires.

AISI, MJ

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In mathematical set theory that is wrong. You may go from one set with an infinite number of elements to another with an infinite numbers of elements, even if the first set is a true subset of the first.

There are infinite number of natural (also called counting) numbers N 0 ),1,2,3,....

"Mathematicians use N or (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-nullhttp://en.wikipedia.org/wiki/Aleph_number#Aleph-null ()".

My red colouring.

There are infinite number of rational numbers Q: n/m where n and m are positive or negative whole numbers.

There are infinite number of irrational numbers I like SQR(2).

There are infinite number of real numbers R = I U(nion) Q

Example:
N subset of Q that is a subset of R.

There are an over countalble number of elements in the Cantor set mentioned in the above post. That is the Cantor Set is non countable. You have to switch you brain to the mathematical part to understand this. Accept the building blocks, axioms and definitions. Doubt the definitions, and you may get a new (most probably inconsistent) theory.

The obeservable universe is 30 billion lightyears and that is definitely finite, though. So it may depend on the elements you are counting in your set or space.

So you did not read about cardinality above?

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Great that you quote that thread.

1. If you stand 1/2 m from the brick wall and you half the distance a finite number of times ( (1/2)^n where n is finite), will you hit the wall?
2. If you kick your foot against the wall hard enough, will it hurt? If it hurts, you have just proved that infintite is a very real concept. You have halved the distance an infinite number of times since: 1/2 x 1/2 x ...... x 1/2 infinite times = 0. (Your foot hit the wall).
3. More precisely: lim (n --> infinity) (1/2)^n = 0

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Physics, biology and chemistry are sciences. Mathematics is not a science as its theory is not the result of scientific experimentation. Mathematics is closer to philosophy than to sciences. You could even say that mathematics is a branch of philosophy,.. like metaphysics.

Jean-Luc

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Kgun. said "In mathematical set theory that is wrong. "

Proving one concept. by conformation with another concept ? ? !

Cardinality? . Show me a cardinality, a "real" One!

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Hmmm, the preceding notwithstanding

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in maths you can create a half, you can divide anything in two. I suspect there is an infinite number of halves.

The first thing a mathematician does (to avoid confusion) is to De-infinitize the half. he creates a boundary -
he starts the half calculation. farmer got 5 cows he sells 1/2 he got 2.5 cows left end of calculation - end of use for that half. the Half 'use' has a start an and end. . It is not infinite.

If there is a failure to create a boundary - 1234567 - to infinity - etc. this Failure creates an infinite number. Maths buffs usually create a boundary. Failure to do so. is the cause of confusion. (foot and the wall) deliberately confusing. There is no infinity. . simply an improper use (belief) of the concept. . everything that exists has a boundary. in math you create the boundary of usage, to make 'an infinity' work.

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I suspect the above is heading towards the right answer Kgun.

The size of an infinity is equal to the boundaries applied to it for the purpose its use is intended.



Well that is my answer today. . (might be different next week)

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P.S.
So If someone said to me "In my mind Gods wisdom is infinite" I have a statement I can accept. This infinity has a clear boundary.

Thank you for asking the question Kgun. It helped me clarify a small muddle..

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Ah, and every boundary implies something beyond!

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Yeah, well, I am not a physicist, so I am out of my league here, but I do know that if you have a powerful enough tool you would be able to measure space between my toe and the wall even as I say "ouch."

Hear, hear!

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How finite is something which isn't finite ? (Well, that's what the question means. I suspect that it's also the answer, isn't it ?)
How wet is completely dry ?

I suspect you can measure the wetness with a percentage of moisture content. o%moisture content.

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I'll need to check that. Is it +0 or -0 ? Did you remember to take away the square of the hypotenuse of the number you first thought of ?
ad infinitum

Another example:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ... Has no finite limit. The sum diverges.

1 + 1/4 + 1/9 + 1/16 + 1/25 + ... Has a finite limit. The sum converges.

Popularising. The elements of the first sum does no go fast enough to zero for the sum to converge. The elements in the second where we square (n*n) the denominator of the elements in the first sum does.

Even if we raise the denominator of the elements in the first sum to the power of s where s > 1 the sum converges, that is it has a finite limit. Do you see the fine distinction? Use an exponent of s=1 as in the first sum and the sum diverges (that is it grows without boundaries). But if n is raised to the power of s > 1 (s >= 1 is a huge error), for instance s = 1.000000000000000000000000000000000000000000001 the sum converges.

Let us take a real life example. There are bonds of infinite life. These bonds has a finite present value, since (according to high school mathematics) it is the sum of a convergent geometric series with a quotient q (0<q<1).

The present value of a bond with infinite life is isomorphic to 1/(1-q).

The same with halving the distance to the wall an infinite number of times. Look at this series (it is the same as the distance one meter from the wall).

1/2 + 1/4 + 1/8 + ... = (1/2)/(1-(1/2)) = (1/2) / (1/2) = 1.

The positive difference between the wall after a finite number of halving the distance can thus be expressed as:

1 - [1/2 + 1/4 + 1/8 + ... + (1/2)^n ] = k > 0. Q.E.D.

If you don't understant this, how can you understand this

The Anatomy of a Search Engine

paper and other more complicated mathematical aspects of the SE's and SEO related topics?

Depending on your answer to this post, there is time for a negative or a positive rep point.

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kgun said.
"If you don't understand this, how can you understand this"

I think most people only wish to understand what we wish to know about. Mostly people are content to know that when they drop a football it falls to the ground. A piece of paper that explains why or how this is possible, or even that if one were to stand on a distant planet how this could simply not happen would be merely a page full of gobbledygook.

most people would simply kick the wall, and ignore the calculations. Some might be mildly amused that someone could think they had the muscle control to move a whole leg one tenth of a millimetre with any expectation of accuracy. Most people accept what is real.

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Tubby, exellent philosophical answer.

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I usualy tell people I am a 'poet'

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A logic party trick. When the conclusion proves that the premise is false it is neither logical, nor correct. Calling something infinite and then proving that it isn't says nothing about infinity, only about the inconsistency of the definition being used.

Happy, I read the above a few times. Are you saying our use the word infinite is inconsistent, or are you saying our use of infinity is inconsistent? or are you saying people are not as consistent as infinity? or are you saying infinity is a totally malleable?

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"The obeservable universe is 30 billion lightyears and that is definitely finite, though. So it may depend on the elements you are counting in your set or space".

"There are bonds of infinite life. These bonds has a finite present value"

While one or both of the above statements may be true (or false), neither represents infinity. Infinity cannot be finite (the words are opposites).

infinity is totally malleable Happy. it is a concept, a tool. It is a tool than can add on (expand) indefinitely, reduce indefinitely. . We use this tool to understand things. There is no Infinity, it is merely a set of rules, you can create rules that conform - to discover answers solutions problems... there is no Infinity it is a concept. Kgun can do what he likes with it. He can use it as a form of mathematical proof. . it works like that. (as long as he follows consistent rules)

But it does not actually exist. it provides proofs.

You can prove Infinity is a concept by simply using it in math to prove it does exist. Then using it in math to prove it does not exist. (proving it to be a concept) . . .

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if a farmer has 100 live cows and ten paddocks, he can use math to work out how he can distribute them evenly.
he calculates 10 cows in each paddock.... he implements his calculations to prove his math.
Yup 10 cows in each paddock

if a farmer has 85 live cows and ten paddocks, he can use math to work out how he can distribute them evenly.
he calculates 8.5 cows in each paddock.... he implements his calculations to prove his math.
oops only 8 cows in each paddock?

Nothing wrong with the calculation? - just unable to prove it . . must be wrong...

the cows are real, math is a concept.

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Note:

1. K finite ==> K/infinite = 0.
2. What about infinity / 2 ?
3. Infinity / 2 a finite number of times?
4. Infinity / 2 an infinite number of times?

There is a concept i mathematics called: completion.

The completion of the real line, R is denoted bar(R).

The Extended real number line

"In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced "positive infinity" and "negative infinity"). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted R or [−∞, +∞]. The affinely extended real number system should be distinguished from the projectively extended real numbers by having two infinities, rather than one.
When the meaning is clear from context, the symbol +∞ is often written simply as ∞."

Note: The bar is lost from R when quoting from Wikipedia.

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And if you're grappling with positive infinity, we have to deal with negative infinity as well...hehe

weren't these the rules?

infinity PLUS or MINUS any positive/negative number = infinity
infinity times (or raised to the power of) any positive number = infinity
infinity times (or raised to the power of) any negative number = negative infinity
anything divided by infinity = 0

Then if we have a sign on infinity, ie. negative, we treat it like we would in algebra, ie.
negative infinity/-2 = positive infinity

and I even believe that the square root of negatve infinity = sqrt(-1)*infinity = i (imaginary) & infinity which = infinity

I could be wrong, this is going back, many, many years now!

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Did you study the above The Extended real number line link? I do not embed links at random in my posts.

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Have to test my memory from time to time!

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But it is only infinity in the same way as the politics of The Soviet Union were Communism, and those of The united States are Democracy. That is to say, a familiar name for something completely different.

There could only be one possible infinity (where would there be room for others ?), and every point, everywhere, would be at its centre. Unimaginable ? Certainly. Measurable ? Certainly not.

if infinity existed, why could there not be infinite infinities? and then - one more....

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kgun said
"Did you study the above The Extended real number line link? I do not embed links at random in my posts."

I rarely leave the WPW page Kgun. For me, you need to summarize what you are saying on the post you make. I have little enough time to play in here. . I do not have time to follow a trail across the net.

A lot of your posts are wasted (shame) Most of us do not understand them! *** chuckling ***

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Where ? Everywhere would already be the centre of the first (only) one.

happy said.
"Where ? Everywhere would already be the centre of the first (only) one."

If there was one center there would need to be be infinite centers . it must by definition be located everywhere. Or it cannot have a center. . or a location. If it has no location it will not exist. If it exists, every property must be infinite. Location is a property.

Infinity is a concept. . I can do what I like with it! so can Kgun.

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I agree. That's why every point is the centre of the only infinity. The distance from anywhere, in any direction, is infinity. I suppose that, if infinity was circular (which it couldn't be, but which at the same time it also couldn't not be) the radius would be half infinity (like half a hole, but on a much grander scale). There must be an infinite number of infinities, to make up the one infinity (they are its building blocks). There could be no other infinities than those contained in the one infinity, because there would be nowhere to put them, except into the one infinity, (where they already are).

So can I.

so can happy!

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Side note:
couldn't not like -- = + ==> could?

The complex plane is topological <==> to the Sphere - Wikipedia, the free encyclopedia. That is the (surface of) the sphere can be projected onto the complex plane.

The surface of the Torus - Wikipedia, the free encyclopedia is also topological equivialent to the plane. Take a sheet of paper, make a cylinder and connect the top to the bottom of the cylinder and you have a torus.

Another side note:

Look at that torus.

1. Take one point, A in the inner circle of the torus and another point, B on the opposite side.
2. Assume that you can only move in 2 dimensions, that is on the surface of the torus (that is topological equivalent to the 2 dimensional plane). That mean that it is the same as moving on the plane. You have to go 180 degrees around the circle to come from point A to point B. So if the radius of the inner circle is 2 meter, you have to move 2 x Pi x 2 / 2 = 2 Pi meter, close to 6.28 meter. If you are free to move in space, the smallest distance is 4 meter.

Conclusion related to the trous. If you go up in dimensions, distance between two points may decrease. So in an infinite dimensional room a distance of 1 meter can be equivalent to 1 light year in a lower dimension

Exercise:

A rope is streched around the globe and second rope is streched 1 meter outside the first rope. How much longer is the outer rope?

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Teaching English again kgun ? You don't know enough about it, so why do you always insist that you know more than others ? You only prove that you don't !

No. Not like could. Like couldn't not. (would be unable not to). It's a simple enough concept. Infinity could not be a circle, because any geometric shape would have a perimeter, which would mean that it was finite. At the same time, when a point is equidistant (i.e. infinity) at all angles it has to be a circle (or a sphere, which is a three dimensional circle). It would be tempting to bring in the fourth dimension, but the inclusion of time would make it an eternity, not an infinity.

Like Tubby, I can't be bothered to check all of your tedious obscure links either. If you have something to say, just say it. If not, don't.

This is a forum, not an exam.

It is called double negation.

I did not say circle. I said surface of a sphere is topological equalivlent to the complex plane.

Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as



Riemann sphere - Wikipedia, the free encyclopedia

It is like "the completion of the real line" that you can read about above.

Visualization:

http://upload.wikimedia.org/wikipedi...tion_in_3D.png







The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection).

By looking at that picture, you should be able to understand this without going into more detail. The links are there for other members.

If I do not remember wrong, the pole point represents the completion.

In words.

Place a ball on the floor and draw a sewing thread around from different positions on the ball. The closer you come to the north pole (see above picture), the larger the circumference of the orbits that the thread traces. On the geograpich north pole, the thread stretches out to infinity on the (complex) plane. The tangent to the sphere on the north pole is horizontal.


Answer:
2 Pi meter.

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kgun: "It is called double negation.

I did not say circle".

Yes, it was a deliberate double negative, not a mistake, as you would understand if you learned English, instead of showing yourself up by attempting to correct other people's.

I "could" tall you what I think of you again, but I won't. "Could" implies choice. If I couldn't not tell you what I think of you, I'd be forced to call you a twerp. There would be no choice.

You say that you didn't say circle. Who said that you did ? The "couldn't not" which you failed to understand, was from this: "...... if infinity was circular (which it couldn't be, but which at the same time it also couldn't not be).....". You see ? You're not the only one posting here, are you ?

Haven't you got a link about that somewhere ?

This point doesn't in fact exist. Simple function 1/x, as x comes in from 'large' negatives it approaches 0 and forms a limit approaching -infinity.

Of course the function picks up in the positive realm again with y coming down from POSITIVE infinity...so you can complete the function but then you're stuck with the fallacy that positive infinity = negative infinity.

and .

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Two non-native English speakers discussing proper English. You have to admit from the point of view of a native English speaker, its a little on the funny side.

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Try this:

"He washed up in his vest and pants"

In English (the language spoken in England), this means that he washed the dishes while dressed in his underwear.

Are you STILL sure that you are a native English speaker, or does it mean something completely different in the language which you speak ?

Yes, the strict high school math say so: K/0 ="Undefined" or "undetermined."

Some persons (see that cite embedded from Wikipedia) defines it as infinity. In Rieman integration theory, integrating over a triangle with base zero, that is a point, the outcome is zero. If you introduce distributions and more generally generalized functions (look at the Dirac Delta function explained in one of my first posts above that is a distribution with all its mass in one point) you can "integrate" over a "triangle with infinite height" (a Dirac puls) with base zero and get integral = area = 1.

Did you know that dicountinous functions can be differentiated by introducing the concept of weak derivative. The discontinuity can not be "too wild". I have seen these function used in investment theory.

So your rules depend on your mathematical topology.


Another well know example:
Ableian groups (a*b=b*a).

The matrix group is a non Abelian group.

You can always conctruct pathological examples. The Cantor set has Lesbesgues measure zero, but there are subsets of the Cantor set that are is non-measureable.

That is there are subsets of a set with measure zero that is not measurable.

If I do not remember wrong mathematical measure theory is heavily used by former Nobel Prize winner in Economics,

Gerard Debreu

or was it

Maurice Allais ?

Link for those interested in Distributions and Generalized functions:

Scroll down to Michael Oberguggenberger, Universitaet Innsbruck:
"Nonlinear SDEs: Colombeau solutions and pathwise limits"

As far as I know he is one of the experts in the world.

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Yes it is

<cite>
The World's Forum for eBusiness Professionals.
</cite>

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What is the point you are trying to make here ?

Are you suggesting that a thread about measuring infinity, coupled with your off-topic, erroneous assertions about the English language, has some bearing on ebusiness professionalism ?

Or is it just yet another unwarranted demonstration of your churlishness ?

I remember once the late Ronnie Scott was being harangued by an oaf with manners like yours, and he responded "Don't drink on an empty head !".

You may find that advice useful (but I doubt it).

THS:

This is the breakroom of "The World's Forum for eBusiness Professionals".

Back to this infinite loop or break?

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I've been to England and have no difficulty understanding them. American English and British English are 99%+ mutually intelligible....are you truly disputing my fluency?

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