Is 0.999... Truly Equal to 1 in the Realm of Infinity?

[lvlastermind]

ram doesn't understand that you can't have 1/infinity because infinity=0 and you can't divide by zero.

 

[ram2048]

Why (and how) are you tacking digits onto .999~?

doesn't matter why and how, but if you must know it's by process of summation where each "step" resolves into one "digit".

9/10 + 9/100 + 9/1000 ...

.999 can never be equal to 1 because it is equal to 999/1000. Of what steps are you speaking?

you knew what i meant.. :P

if you accept Zeno's conjecture as true, within the confines of the problem set forth you can not reach the destination then you also accept that within mathematics .999~ can never equal 1 simply because they are the same problem with a different curve Zeno's is 1/2 and .999~ is 9/10ths.

 

[lvlastermind]

if you accept Zeno's conjecture as true, within the confines of the problem set forth you can not reach the destination then you also accept that within mathematics .999~ can never equal 1 simply because they are the same problem with a different curve Zeno's is 1/2 and .999~ is 9/10ths.

but you will reach your destination with an infinite number of halfs. This is one example of how infinity doesn't act like a real number. Beacause it doesn't have a value but it still has meaning.

 

[Zurtex]

haha my point is proven

not only can you NOT get to infinity, ever. you can NOT get to the largest integer that is not infinity.

We don't add the things one at a time.

 

[ram2048]

but you will reach your destination with an infinite number of halfs. This is one example of how infinity doesn't act like a real number. Beacause it doesn't have a value but it still has meaning.

no this is an example of how calculus uses "infinity" to approximate.

if something cannot logically EVER be something then infinity and forever it will not be it.

how can you possibly reason that the destination is reached? it means that the last step you took wasn't a half-step but a whole one.

i have described earlier how calculus accepts 1/∞ and 2/∞ as the same number because both would be equal to 0.

indeed if 1/∞ = 0 then it is not actually a step at all, so the runner MUST have reached the destination in step 1/2^nint(∞) where nint defines the nearest integer to infinity.

but then we would have the paradox of being able to halve that distance YET AGAIN. such that 1/2^[nint(∞)+1]

and so on... so somehow the conclusion that the destination CAN be reached has to be wrong. or 1/∞ = 0 is wrong. or both are.

i say both personally ;D

 

[Zurtex]

Your argument is wrong.

1/n > 0

Where n is a positive integer. There is NO nearest integer to infinity, that's just silly. Your problem is that you do not seem to be able to grasp a concept of infinity and that it is not on the real number line.

 

[ram2048]

excuse me, but you guys sum n to infinity ALL THE TIME

that means it's on the same number line, it may not be included in your set of reals, but it's still on the same line.

that means somewhere along the line increasingly greater numbers become infinite. if NOT sum n to infinity has NO meaning. like saying sum n to cow or sum n to vacuum cleaner...

the definition of infinity provides a clear relation of "the concept" to known reals such that even though it's not a number it's a function of numbers so much so that it is possible to use it in calculations.

if you're saying a set of reals can never increase to infinity then you're basically accepting that given the infinite number of steps in Zeno's problem, the man will NEVER reach his destination. clear now?

if you understand that much go back to my other post and read the logical explanation on how even at infinity the destination cannot be reached, so it makes no real difference whether a "largest integer" is real or not. the outcome is still the same

 

[ram2048]

sums to infinity are a good approximation

Approximation of what?

approximation of the actual value that would come out of the calculation if you calculated it out longhand. :|

logically

 

[Zurtex]

excuse me, but you guys sum n to infinity ALL THE TIME

that means it's on the same number line, it may not be included in your set of reals, but it's still on the same line.

that means somewhere along the line increasingly greater numbers become infinite. if NOT sum n to infinity has NO meaning. like saying sum n to cow or sum n to vacuum cleaner...

the definition of infinity provides a clear relation of "the concept" to known reals such that even though it's not a number it's a function of numbers so much so that it is possible to use it in calculations.

if you're saying a set of reals can never increase to infinity then you're basically accepting that given the infinite number of steps in Zeno's problem, the man will NEVER reach his destination. clear now?

if you understand that much go back to my other post and read the logical explanation on how even at infinity the destination cannot be reached, so it makes no real difference whether a "largest integer" is real or not. the outcome is still the same

No, your totally wrong as has been proven and shown on this thread many times.

 

[Hurkyl]

doesn't matter why and how, but if you must know it's by process of summation where each "step" resolves into one "digit".

9/10 + 9/100 + 9/1000 ...

This notation looks like an infinite sum; I don't see any "steps".

(To keep things moving)

I'm presuming by "steps" you are first considering 9/10, then 9/10+9/100, then 9/10+9/100+9/1000, and so on. But, of course, none of these are 9/10 + 9/100 + 9/1000 ... (though the limit of these "steps" is)

if you accept Zeno's conjecture as true, within the confines of the problem set forth you can not reach the destination

I have no problem with that. Zeno only considers the motion up to (but not including) reaching the destination. Thus, it would be silly to think that the destination would be reached in the period he analyzes.

then you also accept that within mathematics .999~ can never equal 1 simply because they are the same problem with a different curve Zeno's is 1/2 and .999~ is 9/10ths.

How do you figure? If I do it with 9/10s, then Zeno considers each of these intervals of position [0, 9/10], [9/10, 99/100], [99/100, 999/1000] ...
Putting all of these intervals together yields the interval [0, 1).

excuse me, but you guys sum n to infinity ALL THE TIME

Right, and sums to infinity are defined by

<br /> \sum_{i=1}^{\infty} a_i = \lim_{m \rightarrow \infty} \sum_{i=1}^m a_i<br />
which can be further resolved (by applying the definition of limit)
\sum_{i=1}^{\infty} a_i = L if and only if for every positive \epsilon there exists an integer N such that for any integer m greater than N, we have
<br /> |L - \sum_{i=1}^m a_i| &lt; \epsilon<br /> [/itex]<br /> <br /> Notice, in particular, that this is not logically equivallent to saying that you keep adding terms one by one until you&#039;ve reached an infinite number of terms. (Though, IMHO, it&#039;s for the most part conceptually equivalent)<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> that means it&#039;s on the same number line, it may not be included in your set of reals, but it&#039;s still on the same line. </div> </div> </blockquote><br /> The number line only has real numbers on it, thus it doesn&#039;t have infinity on it. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f61b.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":-p" title="Stick Out Tongue :-p" data-smilie="7"data-shortname=":-p" /><br /> <br /> But, as mentioned before, mathematicians do use an extension of the real numbers which has a positive and negative infinity on each endpoint. But...<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> that means somewhere along the line increasingly greater numbers become infinite. </div> </div> </blockquote><br /> No it doesn&#039;t. A sequence of increasingly greater numbers can <b>converge</b> to infinity, but none of the individual numbers need be infinite... just like a sequence of numbers can converge to zero, but none of them need to be zero. (e.g. 1, -1, 1/2, -1/2, 1/4, -1/4, 1/8, -1/8, ...)<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> if NOT sum n to infinity has NO meaning. </div> </div> </blockquote><br /> I&#039;ll say it again, sum to infinity has this meaning:<br /> <br /> &lt;br /&gt; \sum_{i=1}^{\infty} a_i = \lim_{m \rightarrow \infty} \sum_{i=1}^m a_i&lt;br /&gt;<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> if you&#039;re saying a set of reals can never increase to infinity </div> </div> </blockquote><br /> I&#039;m saying no real number may be infinite, an entirely different statement.<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> you&#039;re basically accepting that given the infinite number of steps in Zeno&#039;s problem, the man will NEVER reach his destination. clear now? </div> </div> </blockquote><br /> I accept that; the destination is reached <i>after</i> the &quot;steps&quot; contemplated by Zeno.<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> approximation of the actual value that would come out of the calculation if you calculated it out longhand. :|<br /> <br /> logically </div> </div> </blockquote><br /> How do you plan to add an infinite number of terms by longhand?

 

[ram2048]

How do you figure? If I do it with 9/10s, then Zeno considers each of these intervals of position [0, 9/10], [9/10, 99/100], [99/100, 999/1000] ...
Putting all of these intervals together yields the interval [0, 1).

how is that different from [0, 1/2], [1/2, 3/4], [3/4, 7/8]...

 

[ram2048]

i'm still not understanding your wording on this

I accept that; the destination is reached after the "steps" contemplated by Zeno

if every step is half the distance (how the problem is defined) then after the "steps" still leaves a whole destination left to cover. if every step is a fraction of 1 whole then somewhere along you believe 1/2 = 1

the only thing i can think of to account for this is your belief that infinity is always equal to itself no matter how you transform it such that 1/2 x ∞ = 1 x ∞.

something like that would make a last step possible but it's not logical at all.

 

[Zurtex]

the only thing i can think of to account for this is your belief that infinity is always equal to itself no matter how you transform it such that 1/2 x ∞ = 1 x ∞.

something like that would make a last step possible but it's not logical at all.

Infinity isn't a real number and doesn't act like that one, so yes that would be correct.

Last possible step? What are you on about?

 

[Hurkyl]

how is that different from [0, 1/2], [1/2, 3/4], [3/4, 7/8]...

Put all of those together and you also get the interval [0, 1).

i'm still not understanding your wording on this

My intent was to state this: the destination is not reached during the sequence of steps considered by Zeno, but that does not imply that the destination cannot be reached at some time that occurs later than the steps considered by Zeno.


However, this reminded me of some of the things I used to point out in Zeno's paradox discussions; the following statement is also true:

At any particular point in time, if it can be said that all of the steps considered by Zeno have occurred, then it is also true that the destination has been reached.

But allow me to emphasize; the destination is not reached during the steps considered by Zeno. In particular, in order to make this statement, I do not make the assumption that one of these steps covers the entire remaining distance. Additionally, I need not use "infinity" anywhere to prove my claim.

the only thing i can think of to account for this is your belief that infinity is always equal to itself no matter how you transform it such that 1/2 x &infin; = 1 x &infin;.

I "believe" it because that is how multiplication by &infin; is defined in the extended real numbers.

something like that would make a last step possible but it's not logical at all.

You mean that it conflicts with your common sense. It is entirely logical because it can be proven rigorously from the definitions and axioms.


But, it shouldn't conflict with common sense. Consider this; when things grow without bound, they approach infinity, right? E.G. the sequence 1, 2, 3, 4, ... approaches infinity, and the sequence 2, 4, 6, 8, ... does as well.

Furthermore, multiplication is a continuous operation. If a sequence approaches a limit, then if I double every number in the sequence, then the limit gets doubled.

Now, 2, 4, 6, 8, ... is the double of the sequence 1, 2, 3, 4, ..., so the limit of 2, 4, 6, 8, ... is double the limit of 1, 2, 3, 4, ..., thus suggesting that &infin; = 2 * &infin; should be correct.

 

[ram2048]

the destination is not reached during the sequence of steps considered by Zeno, but that does not imply that the destination cannot be reached at some time that occurs later than the steps considered by Zeno.

well since zeno considers to infinity, you believe that "beyond infinity" there lies a step such that 1/2 the remaining distance = the whole distance?

what I'm saying is infinity or not you believe you can take 1/2 a distance and it equals the whole distance. basically going back to what Integral said about the smallest thing being a point, what's the 'length" of a point there isn't a length so points cannot be used to fill a "gap" of any distance.

because going backwards in steps you'd also have to have 2x"the length of a point distance left to cover such that you could 'fill' it with your 1x point.

the convergence would look something like...

8 point lengths remaining (move (1/2 x 8) points)
4 point lengths covered

4 point lengths remaining (move (1/2 x 4) points)
2 point lengths covered

2 point lengths remaining (move (1/2 x 2) points)
1 point length covered

1 point length remaining (move (1/2 x 1) point. point is not divisible. move 1 point)
1 point length covered

i don't know if that makes anymore more sense at all, but that last consideration is the one i say wouldn't happen because it would be beyond the confines of the problem.

bear in mind i know a point has no "length" so don't lecture me on that. I'm using it to illustrate a point since it's the smallest conceivable thing.

so using that model, stuff has a limit of divisibility. the end step would break the rules set forth in the problem such that you'd be moving not 1/2 step but a whole step. while we calculate out with infinite steps we still reach that imposed limit caused by the point not being divisible.

but if it WERE divisible it STILL wouldn't be reached because we'd be faced with 1/2 a point 1/4 a point etc etc.

 

[ram2048]

Now, 2, 4, 6, 8, ... is the double of the sequence 1, 2, 3, 4, ..., so the limit of 2, 4, 6, 8, ... is double the limit of 1, 2, 3, 4, ..., thus suggesting that ∞ = 2 * ∞ should be correct

i'm not seeing how that proves infinity=2xinfinity. seems quite the opposite because one sequence is twice the other. how are you saying they are equal?

consider:

when you say something grows you have to apply a rate. if the rate of growth is equal among 2 sequences such that sequence 1 growing double that of sequnce 2. then it's easy to prove that within any given frame of reference in time, sequence one is larger than sequence 2. even at infinity.

1,2,3,4... ...100,101,102... ...1001,1002,1003...
2,4,6,8... ...200,202,204... ...2002,2004,2006...

such that any line cutting perpendicularly into these 2 parallel lines will always yield a number and a number 2xgreater than it.

although if you were to grow things without a rate of time, these number sequences would "exist" as any number within their range at any given time, it would be impossible to accurately assess any relation in value when comparing them. 1 could = 3 or 300,000.

 

[Hurkyl]

you believe that "beyond infinity" there lies a step such that 1/2 the remaining distance = the whole distance?

I believe there is nothing between Zeno's steps and the destination.


Consider that the sequence 2, 4, 6, 8, ... is a subsequence of 1, 2, 3, 4, ...,

  2   4   6   8 ...
1 2 3 4 5 6 7 8 ...

Since the former can be formed by removing terms from the latter, how can the former have a bigger limit?

within any given frame of reference in time

although if you were to grow things without a rate of time,

What does time have to do with anything?

even at infinity.

The sequences aren't even defined at infinity. (unless your "infinity" is a positive integer, in which case it is finite, thus making your use of the term highly misleading)

these number sequences would "exist" as any number within their range at any given time

At any given time, a number sequence is a number sequence. They do not "exist" as anything else.

 

[hello3719]

Ram2048, infinity is a PROCESS and not a number, it only means that you are constantly growing in relation to the certain set of numbers considered.

so you can see that infinity*2 is the same as infinite since in both cases the growing process is present which is what infinity means. growing twice as fast or growing is the same as growing.

 

[ram2048]

I believe there is nothing between Zeno's steps and the destination.

so the sum of that sequence will never equal its limit...

if this is NOT the case you need to explain to me how we're processing along towards the destination with computation and all of a sudden there's no distance left and we're there. There has to be a definable process that gets us there.

Consider that the sequence 2, 4, 6, 8, ... is a subsequence of 1, 2, 3, 4, ...,

  2   4   6   8 ...
1 2 3 4 5 6 7 8 ...

Since the former can be formed by removing terms from the latter, how can the former have a bigger limit?

What does time have to do with anything?

that's why i said growth has to be a function of time or reference in some way "rate" such that the steps can be measured. otherwise if sequence 1 "grew" at a rate 50,000 times slower than sequence 2, it wouldn't matter that they had the same limit or that sequence 1 was "twice the value" of sequence 2, [edit] at the beginning seq 1 would be greater than seq 2 but it would then be overtaken due to superior rate.

The sequences aren't even defined at infinity. (unless your "infinity" is a positive integer, in which case it is finite, thus making your use of the term highly misleading)

that statement was a conclusion stemming from the process. most processes that tend towards infinity have a very predictable output and logical conclusion. with the same growth "rate" and the same start point, the double sequence will hit infinity while the single will only be at 1/2 infinity.

this is the logical conclusion, nevermind that neither will EVER reach infinity,

At any given time, a number sequence is a number sequence. They do not "exist" as anything else.

not so, if no rate is applied to growth it means that it simultaneously exists as any allowable value within its field (start to limit)

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

and as a side note i want to know if this proof works by your math.

x=.999~
10x = 9.999~
10x - x = 9.999~ - .999~
9x = 9

x = 1
.999~ = 1

curious to see if that's allowed or if some crackpot made that up without running it by you guys.

 

[Hurkyl]

Ram2048, infinity is a PROCESS and not a number

That's at least half incorrect. In number systems with a single nonfinite number (or some otherwise especially identifiable nonfinite number), that nonfinite number is often labelled "infinity". Its other uses (such as in \sum_{i=0}^{\infty}) are as a formal symbol to denote something in particular (IMHO it would be misleading to call things so denoted "processes").

so the sum of that sequence will never equal its limit...

The sum of an infinite sequence is, by definition, the limit of the partial sums, so...

if this is NOT the case you need to explain to me how we're processing along towards the destination with computation and all of a sudden there's no distance left and we're there. There has to be a definable process that gets us there.

Zeno's steps cover every point between the start and destination (but not including the destination). If Achilles has completed all of Zeno's steps, where could he be if not at (or past) the destination?

A more analytical answer might go as follows: if p(t) is the position of Achilles at time t, then, according to Zeno's steps, p(0) = 0, p(1/2) = 1/2, p(3/4) = 3/4, p(7/8) = 7/8, et cetera.

Now, lim_{n \rightarrow \infty} (1 - 1/2^n) = 1, and since motion is continuous, we conclude

\begin{align*}<br /> p(1) &amp;= p(lim_{n \rightarrow \infty} (1 - 1/2^n)) \\<br /> &amp;= lim_{n \rightarrow \infty} p(1 - 1/2^n) \\<br /> &amp;= lim_{n \rightarrow \infty} (1 - 1/2^n) \\<br /> &amp;= 1\end{align}

that's why i said growth has to be a function of time or reference in some way "rate" such that the steps can be measured.

That seems necessary to maintain your stance, but I can't see any reason that this is necessary in general.

not so, if no rate is applied to growth it means that it simultaneously exists as any allowable value within its field (start to limit)

Rephrase yourself. It is absurd to say, for instance, 1 = <1, 2, 3, 4, ...>, but I am interpreting you as suggesting that this equality can be correct.

and as a side note i want to know if this proof works by your math.

x=.999~
10x = 9.999~
10x - x = 9.999~ - .999~
9x = 9

x = 1
.999~ = 1

Yes, this is a valid proof.

It is not a valid "first" proof, though, in the sense that this proof depends on the fact that the decimal numbers are a model of the real numbers, so it cannot be used to prove, in the first place, that the decimal numbers are a model of the real numbers.

 

[ram2048]

It is not a valid "first" proof, though, in the sense that this proof depends on the fact that the decimal numbers are a model of the real numbers, so it cannot be used to prove, in the first place, that the decimal numbers are a model of the real numbers.

darn cause i had a good disproof and proof on infinity+1≠infinity from that one :(

oh well

Zeno's steps cover every point between the start and destination (but not including the destination). If Achilles has completed all of Zeno's steps, where could he be if not at (or past) the destination?

he could still be in transit.

\begin{align*}p(1) &amp;= p(lim_{n \rightarrow \infty} (1 - 1/2^n)) \\&amp;= lim_{n \rightarrow \infty} p(1 - 1/2^n) \\&amp;= lim_{n \rightarrow \infty} (1 - 1/2^n) \\&amp;= 1\end{align}

i've always concluded that 1/∞ has a positive value. I've been trying real hard to understand you guys' "something turns into nothing" but it's not getting through :(

Rephrase yourself. It is absurd to say, for instance, 1 = <1, 2, 3, 4, ...>, but I am interpreting you as suggesting that this equality can be correct.

no no, that's not what i meant by it. i meant since it is a defined series of growth, without anything to denote the transition in steps such a rate or time value, it would be anyone of those numbers at the exact instance of transformation.

it's part Schrödinger in a way i guess. but it's kinda veering on a tangent away from the issue at hand hehe.

in any case I'm pretty sure if i can convince myself that 1/∞=0 everything will be clear to me :O

 

[Hurkyl]

he could still be in transit.

If he's still in transit, he has to be someplace between the start and finish.
If he's someplace between the start and finish, then he has not done all of Zeno's steps.

Thus, if he has done all of Zeno's steps, then he's not in transit.



The key thing about the real numbers, that you are not using, is that they are complete. Intuitively, that means they have no "holes"; this is usually stated as follows:

If A and B are nonempty collections of real numbers, such that A lies entirely on the left of B, then there is a number c seperating them.
More precisely,
Let A and B are nonempty sets of real numbers
For all a in A and b in B: a < b
Then, there exists c such that
For all a in A and b in B: a <= c <= b


This can be used to prove the Archmedian property; every number is smaller than some integer (and is thus finite) as follows:

Let A be the set of everything smaller than some integer. (Note that all integers are in A, because n < n+1)
Let B be the set of everything bigger than all integers.
Assume B is nonempty.
Then, there exists some c seperating them.

Thus, a <= c <= b for all a in A and b in B.

Now, c is either bigger than all integers (and thus in B), or it's not (and thus in A).

If c is in A, then it is smaller than some integer. Call it n.
Because c < n, c + 1 < n + 1. Thus, c + 1 is in A.
This is a contradiction because c + 1 > c, but because c + 1 is in A, c+1 <= c. Thus c cannot be in A.

Thus, c is in B.
c - 1 < c, so c - 1 must be in A. c - 1 must be smaller than some integer. Call it n. Because c - 1 < n, we have c < n + 1. However, n + 1 is an integer and is in A, so n + 1 <= c, which is a contradiction.

Thus c cannot be in B.

So our assumption that B is nonempty led to a contradiction: there's a hole between the finite and infinite numbers, but by definition the real numbers have no holes.

Thus, we conclude that there are no infinite numbers; every number is smaller than some integer, and thus the Archmedian property is proven.


Now, we turn back to your question. Suppose 1 / &infin; = e > 0. (whatever 1 / &infin; may happen to mean).

By the Archmedian property, 1 / e is smaller than some integer. Call it n.
Because 1/e < n, e > 1/n.

So, if you insist on maintaining that 1 / &infin; > 0, then there exists an integer n such that 1/n is smaller than 1/&infin;, which is absurd!


Roughly the same argument is used to prove that \lim_{n \rightarrow \infty} 1/2^n = 0.


(More directly, if you maintain that &infin; is a real number, then I could simply apply the Archmedian property to produce an integer bigger than &infin;)

 

[matt grime]

ram, have you considered stepping back, taking a deep breath and admitting to yourself that you don't understand the mathematics involved? that, in the extended numbers, infinity*2=infinity follows from the definitions of that system, and 1/infinity is zero. in the surreals, 1/w is not zero, but w is still not a real number.

As for zeno and trying to understand how you can 'go beyond infinity' as is being talked about, consider trying to learn about transfinite systems, and limit ordinals, you also need to stop thinking in terms of moving from one spot to the next, but in terms of having done all the previous moves (a vague introduction to transfinite induction).

If you tried to understand the limit concept properly instead of presuming you know what it is then none if this nonsense would happen.

 

[ram2048]

So, if you insist on maintaining that 1 / ∞ > 0, then there exists an integer n such that 1/n is smaller than 1/∞, which is absurd!

that's because of the "self defined" upwards limit on infinity itself. BECAUSE you believe it to be all inclusive such that any value added to infinity has no meaning, you're presented with the upwards or downwards limits as such. but this still does NOT prove that the number would equal zero. consider 2 apples next to each other touching. there is no space in between them but they are not the same apple. BECAUSE infinity has a limit such that nothing can be greater than it, no SPACE can exist between 1/∞ and 0, but that doesn't prove they're the same number. infinity is not a number so it doesn't function as one in relation to this "archemidien rule". let's apply infinities "limit" to a real number and see how it works. let's make 500 be the highest possible number there can be. let's now say 1/500. is 1/500 = 0? it is most certainly NOT, yet this is the closest POSSIBLE number to 0 you can get because of the limit imposed.

ram, have you considered stepping back, taking a deep breath and admitting to yourself that you don't understand the mathematics involved? that, in the extended numbers, infinity*2=infinity follows from the definitions of that system, and 1/infinity is zero. in the surreals, 1/w is not zero, but w is still not a real number.

i doubt myself all the time, but until i can find the answers that convey meaning to me in such a way that I'm completely convinced and can properly relay this information to another person that might be having the same problem, i cannot quit.

As for zeno and trying to understand how you can 'go beyond infinity' as is being talked about, consider trying to learn about transfinite systems, and limit ordinals, you also need to stop thinking in terms of moving from one spot to the next, but in terms of having done all the previous moves (a vague introduction to transfinite induction).

well the point of zeno's exercise was that you could NEVER complete all the moves, that's why it was proposed as a paradox...

you want to explain to me then since you know how to do it? none of that fancy math stuff, just logically spell it out for me how it could EVER be possible to cover a whole distance if you can only move in halves of the remaining distance.

 

[russ_watters]

i can goto 5 sites and pull a different definition of real numbers, irrational numbers, and infinity.

and you talk to ME about inconsistency. I can't use your words because your definitions are "Mumbo jumbo"|

Quite frankly, the discussion should have ended here. If you can't even accept that there are specific definitions for these concepts, you can't ever hope to understand the definitions or how to apply them.

If you really do want to understand what we're trying to tell you, this is where you must start.

 

[ram2048]

Quite frankly, the discussion should have ended here. If you can't even accept that there are specific definitions for these concepts, you can't ever hope to understand the definitions or how to apply them.

If you really do want to understand what we're trying to tell you, this is where you must start.

that's BS, russ. i wouldn't have known there were 5 or more definitions for commonly used math terms if i hadn't taken the time to research it and try to find which one was the correct one. i even go through the trouble to annotate myself with "quotes" when I'm unsure as to the correct term. this is possibly the best place to find proper definintions for these terms as if someone brings up the wrong one he is immediately and punitively apprehended... like within the hour. you guys are harsh ;D

and russ, if you have nothing positive to contribute, just don't. really.

 

[matt grime]

these 'different' definitions of real numbers are all equivalent, that is they do the same thing, but that seems beyond your capability to understand.

teach your self transfinite induction, it isn't very hard, any idiot can understand it.

it has been explained to you patiently and accurately what the use of infinity is in mathematics, now it is time for you to go away and think about all that has been written and say what you don't understand, not what you think is worong because your mathematical spohistication is not such that you are at a point to say something is wrong, merely that you don't understand it, and that is a problem with you and not with mathematics.

we are not harsh, you are ignorant. that is not an insult, just a statement of fact.

 

[ram2048]

it has been explained to you patiently and accurately what the use of infinity is in mathematics, now it is time for you to go away and think about all that has been written and say what you don't understand, not what you think is worong because your mathematical spohistication is not such that you are at a point to say something is wrong, merely that you don't understand it, and that is a problem with you and not with mathematics.

so you can't explain it without your fancy math. which leads me to the conclusion that you only believe it because it is popular and fashionable. If you don't understand the math well to be able to rationalize if logically, then all you are doing is parroting back text.

we are not harsh, you are ignorant. that is not an insult, just a statement of fact.

i am not knowledgeable in the ways of "transfusion inducers" or whatever the heck you're talking about, that hasn't stopped me from disproving 4 "accepted" proofs. quite silly how it can all be taken down by an "ignorant" person.

 

[Zurtex]

i am not knowledgeable in the ways of "transfusion inducers" or whatever the heck you're talking about, that hasn't stopped me from disproving 4 "accepted" proofs. quite silly how it can all be taken down by an "ignorant" person.

That is your problem, you automatically assume you are right, you are unwilling to learn mathematics, you have not actually disproved anything and that makes you ignorant.

 

[chroot]

When an intelligent man argues at length with a fool, it becomes difficult to tell them apart.

- Warren