Inifinity Question

[Paree]

I just have a question regarding infinity. What I'm saying is probably a common misunderstanding, but I would just like somebody to clear it up for me. Here it is.

If, for example, we took the geometric series: Un=0.5x0.5^(n-1), and sum it to infinity, that is, Lim n=>(Un=0.5x0.5^(n-1)), which=0.5/0.5 = 1.
Doesn't this imply that 1 is a product of an infinite number of finite numbers? If so, does this also imply 1 is infinite (and 2 and 3 etc)?

To put it in context, say, for example, you put your hand 1 "meter" apart, then moved them 0.5 meters closer, then 0.25, then 0.125 and so on. You would need to pass an infinite amount of space until your hands touched, yet you would have only passed through 1 meter. Another example could be with time; in order for you to pass from 1 second to 2 seconds, an infinite amount of time must go by, yet still only 1 second.

Could someone clear this up, because I'm sure that I'm wrong?

[HallsofIvy]

"I just have a question regarding infinity. What I'm saying is probably a common misunderstanding, but I would just like somebody to clear it up for me. Here it is.
If, for example, we took the geometric series: Un=0.5x0.5^(n-1), and sum it to infinity, that is, Lim n=>(Un=0.5x0.5^(n-1)), which=0.5/0.5 = 1.
Doesn't this imply that 1 is a product of an infinite number of finite numbers? If so, does this also imply 1 is infinite (and 2 and 3 etc)?"

I'm sorry, why a "product"? It seems to me you just showed that it was a SUM of an infinite number of finite numbers. That is true but does not show that 1 itself is in any sense "infinite". It is quite possible for the sum of an infinite series of numbers to be finite. (It is necessary that the numbers themselves "go to" 0 in the limit, very fast.) Much of calculus deals with that.

No, neither of those follows. In a certain sense there are an infinite number of POINTS between 0 and 1 but since each point takes up no space, an "infinite amount of space" does not follow. And you give no justification for there being "an infinite amout of time" within 1 second.


"To put it in context, say, for example, you put your hand 1 "meter" apart, then moved them 0.5 meters closer, then 0.25, then 0.125 and so on. You would need to pass an infinite amount of space until your hands touched, yet you would have only passed through 1 meter. Another example could be with time; in order for you to pass from 1 second to 2 seconds, an infinite amount of time must go by, yet still only 1 second."

[Paree]

Ah sorry, I did mean sum...But yes you have cleared this up for me, I knew what I was thinking had to be wrong. With the time example I was using the same justification as before, that is: 0.5 second pass then 0.25 etc...

[OfficeShredder]

Before the concept of infinite summations was developed, that very paradox was used to show that mathematics didn't reflect the real world. Now we know that if you spend .5 seconds, .25 seconds, .125 seconds... all in a row, it's just like spending 1 second. How long do you actually expect 1/2^1000 seconds to last anway? :D

[Epsilon=One]

It doesn't appear from the prior responses on this thread that anyone has a firm grasp of the meaning of infinity.

This is not too surprising because Georg Cantor, who might be called the father of infinity, admittedly never understood the concept; as is the case with all subsequent mathematicians; many of whom have the audacity of compiling more than forty different definitions of infinity . . . which is a/the singularity.

There is much confusion between the connotations of infinite and Infinity (http://www.CQthus.com/PT/I) that is only increased when considering the infinitesimal.

The unit between integers considered as "One" is little related to the infinite or the infinitesimal. However, "One" as a singularity is heuristically, symbolically, and literally related to Infinity (http://www.CQthus.com/PT/I).

See the following four posts for some more insight:1. Infinity (http://www.CQthus.com/PT/I);
2. Quantum constants' relation to Natural integers (http://www.CQthus.com/PT/QC);
3. The Dynamic Nature of Dimensionless points,
.........dimensionless spheres, and their separation (http://www.CQthus.com/PT/DS); and
4. The radius of Infinity (ROI) is knowable. (http://www.CQthus.com/PT/ROI)

[OfficeShredder]

This is not too surprising because Georg Cantor, who might be called the father of infinity, admittedly never understood the concept;

Father of infinity? That's a bit of a stretch

as is the case with all subsequent mathematicians; many of whom have the audacity of compiling more than forty different definitions of infinity . . . which is a/the singularity.

Of course you need different definitions depending on the context.... in case you haven't noticed, infinity is never defined as "something incredibly huge", but instead is commonly used to denote that whatever is written down is larger than anything finite. The definitions are quite satisfactory in fact.

[Epsilon=One]

Father of infinity? That's a bit of a stretchWhy would you say this? Who did more original work on the subject at an earlier date than Georg Cantor?

...you need different definitions depending on the context....This would be true for any word/concept. Usually, infinity is considered as a singularity; mathematicians generally consider infinity as a limit.

If infinity is a singularity, or a limit that is unreachable, then, no more than one rather "tight" definition is required.

Mathematicians tend to consider large numbers, such as the sequence of integers, etc. as an infinity. Of course, this is ridiculous; as no number is a singularity or beyond reach of the preceding number.

No definition of infinity can be applied to anything that exists; thus, again, the definition is limited to a singularity, which makes the numerous definitions of mathematics a bit ludicrous . . . unless, of course, you are a mathematician.

...infinity is never defined as "something incredibly huge", but instead is commonly used to denote that whatever is written down is larger than anything finite.Because you use the word "larger" you have refuted your argument.

You do infer something beyond the finite, which is not without merit; and thus, you indicate a singularity, which is precisely my argument.

I have never met a mathematician that has been able to quickly grasp the simple concept of Infinity (http://www.CQthus.com/PT/I). Yet, children seem to have little difficulty with the concept as they have not been confusingly impressed.

The definitions are quite satisfactory in fact.This is not so . . . "in fact." The mindset of mathematicians may be fine for manipulating the various forms of infinity that they propose; however, this mindset is quite deleterious for a physicist's understanding of Reality (http://www.CQthus.com/PT/R). And, physicists that are often ill-equipped for philosophical logic beyond that of mathematics are ill-equipped to define the Reality (http://www.CQthus.com/PT/R) that they pursue.

Without a clear, simple understanding of Infinity (http://www.CQthus.com/PT/I) there can never be a clear simple understanding of the environment that we find ourselves within.

[OfficeShredder]

Why would you say this? Who did more original work on the subject at an earlier date than Georg Cantor?

On what? Infinity? Euler did an incredible amount of work on limits, infinite series, etc.

This would be true for any word/concept. Usually, infinity is considered as a singularity; mathematicians generally consider infinity as a limit.

Infinity is not usually considered a singularity, rather, singularities are usually considered infinite.

If infinity is a singularity, or a limit that is unreachable, then, no more than one rather "tight" definition is required.

So what would your tight definition be? It should be noted that there is a very specific definition of infinity as a limit.

Mathematicians tend to consider large numbers, such as the sequence of integers, etc. as an infinity. Of course, this is ridiculous; as no number is a singularity or beyond reach of the preceding number.

The sequence of integers isn't a number? If something is larger than every natural number, what would you call it?

No definition of infinity can be applied to anything that exists; thus, again, the definition is limited to a singularity, which makes the numerous definitions of mathematics a bit ludicrous . . . unless, of course, you are a mathematician.

I don't understand the point you're making here. The definition of a limit going to infinity is often used in dynamics to show that a system "explodes" (of course, explode being a colloquial use here, since the proper term is that it goes to infinity)

You do infer something beyond the infinite, which is not without merit; and thus, you indicate a singularity, which is precisely my argument.

When did I say there was something beyond the infinite?

I have never met a mathematician that has been able to quickly grasp the simple concept of Infinity (http://www.CQthus.com/PT/I). Yet, children seem to have little difficulty with the concept as they have not been confusingly impressed.

I have never met a mature adult that has been able to quickly grasp the simple concept of Santa Claus being real, yet children seem to have little difficulty with the concept as they have not been confusingly impressed.

Very nice

[Epsilon=One]

On what? Infinity? Euler did an incredible amount of work on limits, infinite series, etc.You can always argue accomplishments and priority. I consider Euler as one of the greatest mathematicians that ever lived . . . entirely above Cantor. (I'm a Euler fan as was my native Swiss grandfather.) And, Euler certainly was aware of limits well before Cantor was born.

However, Cantor and infinity are almost synonymous. See: Counting to Infinity (http://http://scidiv.bcc.ctc.edu/Math/infinity.html). There are many books and references that credit Cantor with the "invention" of a wide variety of infinities; particularly concerning sets.

I may have been a bit over enthusiastic concerning Cantor's standing; though I'm sure there are many leading mathematicians that would concur.

Infinity is not usually considered a singularity, rather, singularities are usually considered infinite.This is why I’m so adamant. Logically, a proper connotation for the word singularity would rule out the possibility that a singularity can be plural; as you use the concept. You may define anything as you wish; however, logic must be considered when drawing a definition. I draw the line that a singularity means exactly what it implies.

So what would your tight definition be? It should be noted that there is a very specific definition of infinity as a limit.I have no argument with “a very specific definition of infinity as a limit.” My argument is with the forty some definitions that mathematicians consider for the concept of infinity. Or, that mathematicians apply fine and subtle differences to their definitions.

Physically, infinity as a singularity must be a function of motion.

It appears from your post that you have not bothered to read my definition of Infinity (http://www.CQthus.com/PT/I). It is this definition that I am arguing. And, I always indicate it with font and case so there will be no misunderstanding.

The sequence of integers isn't a number? If something is larger than every natural number, what would you call it?I would not “call it” Infinity (http://www.CQthus.com/PT/I). I would have no objection to referring to a large unending sequence as: “infinite”; however, I would prefer a “large unending sequence.” The definition hinges upon whether the “something” you refer to exists or not.

I don't understand the point you're making here. The definition of a limit going to infinity is often used in dynamics to show that a system "explodes" (of course, explode being a colloquial use here, since the proper term is that it goes to infinity)I understand that you don’t understand. Mathematicians seldom do; I understand. My point is a singularity is single; a singularity does not exist; and, a singularity is defined by the absence of dimensions, thus motion. Most simply: a singularity/Infinity does not exist.

When did I say there was something beyond the infinite?You did not. I misquoted you or made a typo. I properly should have stated “finite” rather than “infinite.” I have since corrected paragraph seven of said post.

Thank you for pointing out my error.

I have never met a mature adult that has been able to quickly grasp the simple concept of Santa Claus being real, yet children seem to have little difficulty with the concept as they have not been confusingly impressed.I dislike analogies and anecdotes; and, probably should not have used it. However, for my intent it was quite apropos.

I do not feel so strongly about your Santa Claus rebuttal. But, it did bring a warm “et tu” smile.