# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

ATTENTION MODERATORS/ADMIN. Reporting the troll has proved ineffective. Tach continues to dishonestly evade. The only recourse is to bring this to your attention in open forum: Here is the exchange so far...

Undefined said:
Hi Tach.

This was a question put to you pursuant to YOUR 'suggestion' to rpenner that you could 'share' the endpoint. I therefore asked YOU as follows::
Undefined said:
rpenner said:
Those points have measure zero, so they really don't count, but a simple procedure of assigning each point to it's clockwise-adjacent arc will split the circle into N congruent parts.

"Sharing" the endpoints by congruent arcs accomplishes the same result.

Tach, can you take us all through the reality process of 'sharing' a DIMENSIONLESS 'endpoint' in reality context? Thanks.

PS: Just making abstract axiomatic assumptions is not what is asked for. What is asked for is the reality process of effecting such a 'sharing' of such an 'endpoint' in this reality context. Thanks.

Now your answer below is an EVASION pure and simple...

As soon as you manage to solve the 8-th grade exercise of dividing a circle in three equal parts using a ruler and a compass. You still owe the solution to the exercise.

Please remember that you have been banned before for such dishonest evasion etc tactics.

So please answer the question asked because YOU made, to rpenner as above quoted, the suggestion of 'sharing' the 'endpoint'. Please show how this 'sharing' of a DIMENSIONLESS (as rpenner confirmed) abstract 'point' is done, precisely, in reality.

Any more dishonest evasions and you will be reported. Thankyou.

If anyone is evasive, that one is you. I have given you the simple exercise of dividing a circle in 3quite a long time ago. It's been 4 days and you still don't know how to solve it. You have a lot of difficulty with an exercise that is soved routinely by 8-th graders, yet you pretend (like chinglu) to be discussing higher level concepts.

Tach, you chose to dishonestly evade again. You have been banned previously for that. You have been reported in this instance also.

Please stop evading now, and answer properly and honestly OR admit you don't know what you meant when you suggested 'sharing' a dimensionless 'endpoint'. Thankyou.

Edit:

Tach: If you continue to evade instead of either answering the question OR admitting you don't know, then you are vulnerable to being banned for evading. How many infractions is that against you now?

Tach, you chose to dishonestly evade again.

Not at all, you are still stuck on solving the simple exercise that would teach you how to divide a circle in 3 equal parts. The fact that you cannot do it blocks your ability of learning that it can be done (actually, it has been done for the last 2000 years. Remedial class in 8-th grade geometry is highly recommended for you. Oh, and please stop trolling with your silly demand as to how to "divide a point".

ATTENTION MODERATORS/ADMIN. Reporting the troll has proved ineffective. Tach continues to dishonestly evade. The only recourse is to bring this to your attention in open forum: Here is the exchange so far...

Undefined said:
Hi Tach.

This was a question put to you pursuant to YOUR 'suggestion' to rpenner that you could 'share' the endpoint. I therefore asked YOU as follows::
Undefined said:
rpenner said:
Those points have measure zero, so they really don't count, but a simple procedure of assigning each point to it's clockwise-adjacent arc will split the circle into N congruent parts.

"Sharing" the endpoints by congruent arcs accomplishes the same result.

Tach, can you take us all through the reality process of 'sharing' a DIMENSIONLESS 'endpoint' in reality context? Thanks.

PS: Just making abstract axiomatic assumptions is not what is asked for. What is asked for is the reality process of effecting such a 'sharing' of such an 'endpoint' in this reality context. Thanks.

Now your answer below is an EVASION pure and simple...

As soon as you manage to solve the 8-th grade exercise of dividing a circle in three equal parts using a ruler and a compass. You still owe the solution to the exercise.

Please remember that you have been banned before for such dishonest evasion etc tactics.

So please answer the question asked because YOU made, to rpenner as above quoted, the suggestion of 'sharing' the 'endpoint'. Please show how this 'sharing' of a DIMENSIONLESS (as rpenner confirmed) abstract 'point' is done, precisely, in reality.

Any more dishonest evasions and you will be reported. Thankyou.

If anyone is evasive, that one is you. I have given you the simple exercise of dividing a circle in 3quite a long time ago. It's been 4 days and you still don't know how to solve it. You have a lot of difficulty with an exercise that is soved routinely by 8-th graders, yet you pretend (like chinglu) to be discussing higher level concepts.

Tach, you chose to dishonestly evade again. You have been banned previously for that. You have been reported in this instance also.

Not at all, you are still stuck on solving the simple exercise that would teach you how to divide a circle in 3 equal parts. The fact that you cannot do it blocks your ability of learning that it can be done (actually, it has been done for the last 2000 years. Remedial class in 8-th grade geometry is highly recommended for you. Oh, and please stop trolling with your silly demand as to how to "divide a point".

Again you have chosen to dishonestly evade answering.

You are asked to explain your suggestion to rpenner to 'share' a dimensionless 'endpoint'. The exercise in reality context, not in mathematical abstraction, is what the question asked requires you to explain about your 'suggestion'. Nothing else is involved.

Please stop evading now, and answer properly and honestly OR admit you don't know what you meant when you suggested 'sharing' a dimensionless 'endpoint'. Thankyou.

Edit:

Tach: If you continue to evade instead of either answering the question OR admitting you don't know, then you are vulnerable to being banned for evading. How many infractions is that against you now?

The ONLY improvements possible to existing axioms would be to remove any ambiguities, of which AFAIK there are none. You could remove an axiom and have different consequences follow, or add new ones. There is a whole lot of literature on the geometry that result when the axiom that parallel line do not intersect is removed to allow that they can.

With in the very useful normal set of axioms new concepts, like imaginary numbers, have been well defined and are quite useful in the real world, especially when describing the behavior of voltage and current relationship in circuits with reactance; however there is no claim within the mathematics that imaginary numbers will have any real world application. Some electrical engineers have discovered that they do (and that the response of linear circuits to complex wave forms is most easily achieved by expressing those forms in terms of their Fourier components). Likewise quantum physics makes great use of mathematics, but mathematics it self is a CLOSED TAUTOLOGY and like all such closed tautologies can make NO CLAIMS of application outside its closed domain. There are whole fields of mathematics that AFAIK are of interest only to mathematicians. (Perhaps parts of number theory are totally without any real world application, but large prime numbers are very valuable for making nearly unbreakable codes.)

If you apply some math to a real world problem and it is helpful, fine, but if in some situations doing that leads to a paradox or confusion, the fault is with you, not mathematics. Mathematicians warned you up front that it was a CLOSED TAUTOLOGY, based on an arbitrary set of postulated axioms. For example 41/333 is completely well defined both in the real world and in mathematics when also expressed as a decimal too. I. e. as the infinitely long sequence 0.123123123123... but the real world has problem with 41/333 expressed that way as all calculations are finite.

SUMMARY:
It is non-sense to speak of "CURRENT AXIOMATIC SUFFICIENCY/DEFICIENCY." (Sufficiency or deficiency for what?) Yes, the logical deductions from the current, most commonly postulated axioms can be misapplied.
All I have seen posted here faulting mathematics is ignorance and wasted time. If they don't like the results the current most common set of axiom produces, then suggest changes in the current set, either by adding new axioms or deleting one or more.
well said... IMO
It is indeed a common problem when some mix the two paradigms, mathematics and "real world". I have been and in some ways still are guilty of such an approach but am endeavoring to re-organize the way I think so as to accommodate it better.
Most of the angst seen on this board is associated with just this issue IMO.
Where some mathematicians believe they are expressing a real world view and some non-mathematicians believe they "should" express a real world view. aahgg! confusion reigns supreme!!

It is indeed a common problem when some mix the two paradigms, mathematics and "real world". I have been and in some ways still are guilty of such an approach but am endeavoring to re-organize the way I think so as to accommodate it better.

Math is math, physics is physics. It's as simple as that.

It's as if you and I were out to lunch and I suddenly said to you, "You haven't gone ten yards so you can't have a new first down."

You might say to me: "Yes, but we're eating lunch, NOT PLAYING FOOTBALL."

[American football, of course. I realize there's an international audience here and I don't mean to confuse anyone ]

It's the same thing in these .999... = 1 discussions. That's a basic, established fact in mathematics. But in the real world it's meaningless since there are no infinite sequences of real numbers in the physical world.

So if I say that .999... = 1 and someone says, Well, you can't really show me that in the real world, the answer is Duh, I'm not talking about the real world, I'm talking about mathematics.

It's true that math is incredibly useful in the real world, but math itself is an abstract mental construct.

ps I have no idea why I'm bothering to belabor this point, which I'm sure has already been made dozens of times in this thread and millions of times in similar threads all over the Internet.

Math is math, physics is physics. It's as simple as that.

It's as if you and I were out to lunch and I suddenly said to you, "You haven't gone ten yards so you can't have a new first down."

You might say to me: "Yes, but we're eating lunch, NOT PLAYING FOOTBALL."

[American football, of course. I realize there's an international audience here and I don't mean to confuse anyone ]

It's the same thing in these .999... = 1 discussions. That's a basic, established fact in mathematics. But in the real world it's meaningless since there are no infinite sequences of real numbers in the physical world.

So if I say that .999... = 1 and someone says, Well, you can't really show me that in the real world, the answer is Duh, I'm not talking about the real world, I'm talking about mathematics.

It's true that math is incredibly useful in the real world, but math itself is an abstract mental construct.

ps I have no idea why I'm bothering to belabor this point, which I'm sure has already been made dozens of times in this thread and millions of times in similar threads all over the Internet.
What you are saying is quite correct however..
• What is the point of maths?
• What is it's objectives?
• Where is it applied when it's real world application is essential?

Most would conclude that physics as an umbrella field is the answer to the above.
• Why is the use or misuse of the "axiom of infinity" critical to the issue of 0.999...=1
• Why is the use or misuse of the "axiom of infinity" critical to the way physics looks at the universe?

If the use of the "axiom of infinity" is essential to the solution 0.999... = 1

Then is it not important to understand why it is therefore so essential that infinity be limited to terminating at zero and granted a quasi finite status?

If the use of the "axiom of infinity" is essential to the solution 0.999... = 1

Then is it not important to understand why it is therefore so essential that infinity be limited to terminating at zero and granted a quasi finite status?

You are absolutely right that the Axiom of Infinity is essential to the theory of infinite series. And that's why math differs from physics. Nobody's ever seen an infinite set in the real world and it's highly questionable as to whether one exists. We have no basis in experience for making such an axiom.

Rather, the Axiom of Infinity is based on the intuition that we all have of the counting numbers 1, 2, 3, 4, 5, ... The Axiom of Infinity basically says that there exists a set that models our intuition of the counting numbers ... bearing in mind that "set" is an undefined term! We don't even know what a set is, but we assume there's one that looks like the counting numbers.

Once you grant the Axiom of Infinity then you can model the counting numbers within set theory, and then you can model the integers, rationals, and reals; and then you can develop the theory of infinite series, and then you can prove that .999... = 1.

It's purely an exercise in fictional, abstract math.

But physicists and engineers use infinite series all the time. Digital signal processing, which underlies all computer communications including the Internet, relies on Fourier series. It's perhaps interesting that Cantor was actually studying the discontinuities of trigonometric series when he was led to the discovery of set theory.

How can it be that starting from an axiom that's obviously false about the real world, leads us to a theory that's so useful in the real world? That is a legitimate mystery.

How can it be that starting from an axiom that's obviously false about the real world, leads us to a theory that's so useful in the real world? That is a legitimate mystery.
Why do say that?

Zeno of Elea had no problems describing the use of infinity in the real world, as did Archimedes . I would argue, quite successfully too, I might add, that infinity does indeed exist as a reality but have never seen arbitrary limits placed up on it in reality, as is the case with 0.999... = 1.
Does infinity define the math or does math define the infinite?

Only have to look at a sphere or a ball and infinity is in view.

The human mind, due to it's inherent fears of chaos, needs to be able to manage the infinite and thus a finite method was developed. IMO

For example:

Zeno of Elea and no doubt other well known ancient per-Aristotle Greeks easily recognized that you could expand something in to an infinite volume and also contract something infinitely as well.

ie. 0.999... = 1
or 9999.... = ? [no decimal place]
Allegation:
Zeno recognized a paradox existed and since then maths has been trying to resolve it. They did that by placing an arbitrary limit on infinity.. and I think I hear an ancient Greek voice from the distance, in Elea, yelling "δεν είναι δίκαιο" or "Not fair"..."You cheated!" when contemporary math has Achilles beating [or calling a draw] the tortoise to his own position...thus claiming the paradox was never a paradox.
and Zeno was not talking about Mathematics he was talking about an observation he and others could witness in reality.
"Probably when the ancient Greeks raced each other they probably joked about how no one could beat any one to their own position in a race - ["I always win and you always loose" type kidding around] and Zeno just formalized it into his famous paradox.

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seriously,
if 0.999... =1
then what does 9999... = ? [no decimal places]

ironically 999... = 1(0) could be a rather profound answer... [1 being "whole", complete, totality, universal sum etc]

"the infinite oneness of it all...." [chuckle]

seriously,
if 0.999... =1
then what does 9999... = ? [no decimal places]

ironically 999... = 1(0) could be a rather profound answer... [1 being "whole", complete, totality, universal sum etc]

"the infinite oneness of it all...." [chuckle]

0.999... = 1 because we are talking about the real numbers and
$$0.999... \equiv 0.9 + 0.09 + 0.009 + \dots \equiv \sum_{k=1}^{\infty} \frac{9}{10^k} \equiv \lim_{n\to\infty} \sum_{k=1}^{n} \frac{9}{10^k} = \lim_{n\to\infty} \left( 1 - 10^{-n} \right) = 1 - \lim_{n\to\infty} 10^{-n} = 1 - 0 = 1$$
where $$\equiv$$ denotes equivalence by basic definition and $$=$$ denotes application of the axioms of the real numbers.

...999 is however not a real number because
$$...999 \equiv 9 + 90 + 900 + \dots \equiv \sum_{k=0}^{\infty} 9 \times 10^k \equiv \lim_{n\to\infty} \sum_{k=1}^{n} 9 \times 10^k$$
because this limit does not exist in the real number system. It is not bounded in magnitude.

There is a math system where ...999 has meaning, which is called the p-adic numbers. Wikipedia, Wolfram Mathworld
The p-adic numbers are useful in a different way than the reals -- both extend the rationals but only the reals keep the same idea of magnitude as we mean in geometry. In addition, each different prime p gives you a different p-adic number system. In fact in a p-adic expansion of a number the further you move to the left, the less significant your digits become.

For example the difference (p-adic metric) between 100 and 2 in the base 7 p-adic numbers is $$\left| 100 - 2 \right|_7 = \left| 202_7 - 2_7 \right|_7 \left| 200_7 \right|_7 = \frac{1}{49}$$.

And this leads to a number system where $$...013201320132_7 = \frac{-3}{100}$$ because $$...013201320132_7 \times 202_7 + 3 = ...026402640264_7 + ...640264026400_7 + 3_7 = ...000000000000_7$$.

For example $$\pm \sqrt{2} = ...266421216213_7, \; \mp \sqrt{2} = ...400245450454_7$$ in base 7 p-adic numbers.
You can kind of see how this works in ordinary arithmetic modulo 7:
$$\begin{array}{rcrcrcr} 3_7^2 & = & 1\underline{2}_7 & \quad & 4_7^2 & = & 2\underline{2}_7 \\ 13_7^2 & = & 20\underline{2}_7 & \quad & 54_7^2 & = & 430\underline{2}_7 \\ 213_7^2 & = & 4600\underline{2}_7 & \quad & 454_7^2 & = & 32000\underline{2}_7 \\ \vdots & & \vdots & \quad & \vdots & & \vdots \\ 266421216213_7^2 & = & 116446165651000000000000\underline{2}_7 & \quad & 400245450454_7^2 & = & 22030343022200000000000\underline{2}_7 \\ \vdots & & \vdots & \quad & \vdots & & \vdots \end{array}$$

But while ...999 makes a kind of sense because displayed digits all have fixed meanings, the 999... doesn't begin to have meaning since the meaning of the first 9 is undefined.

because this limit does not exist in the real number system. It is not bounded in magnitude.
So whilst 0.9(9) is limited, 9(9)... isn't. [if I understand you correctly]

thanks rpenner very interesting....

In a sense the idea of asymmetrical use of the limit makes sense.
let us presume that no limits are involved for a moment.

If we take a sphere and reduce it infinitely we must assume that the reduction is contained with in the sphere [IN] and is infinite yet it leads to a single unresolvable point.

However if we expand the sphere outwards [out] it must lead to encompass all possible points [ not a single point ] and is like wise unresolvable.
so infinity in this context is asymmetrical...
Physically limited [Inwards towards 1 point] and physically unlimited [Outwards from zero] towards encompassing all possible points.

This is possibly one of the many reasons the limits were developed the way they were...? Perhaps?

IMO very interesting indeed!

The circle has an area of 1 m^2. You divide it into 3 parts. What is the area of each part and what is the remainder left over that has to be divided by 3?

You can consider the area of a circle as pi*r^2, where r is the radius. You can divide the circle geometrically into three equal parts and there will be no remainder left.

....

Consider that such 'division' cannot be accomplished in reality, because:

There is no way in mathematics/reality to treat the CENTRE POINT which in handsa's 'method' is the "origin" point FOR the 'process' of division he suggested would do the trick. ...

Going by your logic, you cannot divide a circle into two or four equal parts also, because in these cases also the centre point will remain in the origin; just like the case for dividing the circle into three equal parts.

deleted as trivial

Just to help understand how science perceives infinity the following video was posted to the other thread.
The video is about a fictional infinite hotel called Hilbert's Hotel.

Complaint:
the claim that the hotel manager can suddenly just free up rooms in a fully occupied infinite room hotel is totally unsupported in the video.
I wonder how this could be possible?
If all infinite rooms are occupied then where do the free rooms come from?
Are we not simply making a false claim that the hotel was infinitely occupied to begin with? [in the video]

Complaint:
the claim that the hotel manager can suddenly just free up rooms in a fully occupied infinite room hotel is totally unsupported in the video.
I wonder how this could be possible?
The video says "Hilbert's answer is just to make each guest shift along one."
Is there any guest that can't move to the next room?
So now room number 1 is free, right?
If all infinite rooms are occupied then where do the free rooms come from?
They're the same rooms as before, but we free up rooms by changing the relationship of the infinite passengers to the infinite rooms.
Are we not simply making a false claim that the hotel was infinitely occupied to begin with? [in the video]
No. What rooms were vacant?

The video says "Hilbert's answer is just to make each guest shift along one."
Is there any guest that can't move to the next room?

They're the same rooms as before, but we free up rooms by changing the relationship of the infinite passengers to the infinite rooms.
how can you just shift one passenger to another room when they are all occupied infinitely?
On what logical grounds can you say infinity some how implies an open ended situation when you have already claimed all infinite rooms are full?

one possible slant:

The manager says to the incoming guests, "You can't be looking for a room, as you already have one...after all are you not a part of the infinite guests to begin with? So what room do you already have?

the rational being explored is:
Infinity may be "never ending" but it is always a complete series [ with in a given scenario ].
In this case we have an HOTEL with infinite rooms and infinite guests. [ a "complete" yet "never ending" scenario ]
or;
Since when has an infinite number of guests not included all possible guests?

No. What rooms were vacant?
this also suggests a contradiction.
We can have an infinite number of rooms but a less than infinite number of guest....

"We just move them along one..." implies to me that whilst the rooms are infinite the guests aren't.

(R1,G1) + (R2,G2) + (R3,G3) ....
whereby the rooms and guest are directly associated infinitely.
All possible rooms and guests are accounted for. [in this single HOTEL]

Going by your logic, you cannot divide a circle into two or four equal parts also, because in these cases also the centre point will remain in the origin; just like the case for dividing the circle into three equal parts.

Hence the question in all such exercises offered so far to 'sector/halve' MD's real disc:

"How does one 'share' a central point that is 'dimensionless'; or a middle line that has 'no dimension other than length' ?"

Can you or anyone else answer that for me? I would be very interested to see how you and others would do this 'sharing' process in reality. Thanks.