Is 1 = 0.999... ? Really?

[James_S_Saint]

@Carleas

James, are you just articulating Zeno’s paradox?

Suppose I want to go 1 mile. To get there, I have to go 90% of the way. Then I have to 90% of the remaining distance, and 90% of that distance, and 90% of that distance. If I go the whole mile, how many times did I go 90% of the remaining distance? An infinite number of times.

It seems like you want to say that, once I’ve gone 90% of the remaining distance an infinite number of times, there’s still a remaining infinitesimal distance to go. Are you positing some indivisible quantum of number-ness that isn’t part of the infinite number of times we go 90% of the remaining distance? Why can’t we go 90% of that distance?

No, no. Zeno’s is merely a high school calculus puzzle. Calculus is about the relation between two different infinitesimal variables. In this case, both reducing at the exact same rate leaving a constant relation between them, speed.

For every distance traveled, x, there is an associated time, t.
As each x gets shorter, the time to traverse, t, it gets proportionally shorter.
Whether specific or infinitesimal, the speed is the ratio between the two, x/t.

And YOU are proposing that there is an indivisible quantum of “numberless-ness” (ask yourself why you consistently accuse of your own guilt). YOU are proposing that eventually the infinite series runs down to finally get to zero. I am saying that it never runs out, period. It never comes to a conclusion or resolve. And that is part of why it cannot be equal to an actual number. It ISN’T a number, but an endless series of digits.

No “number” that ends with that ellipsis, “…” is an actual real number. The ellipsis is telling you that no number can ever be achieved, because by definition, there is no end to the implied operation, and thus there can be no eventual value. b “0.999…” is unbounded. But “1.000” is bounded.[/b]

@wtf

@James S Saint

Note that my proofs 3 and 4 refer to the fact that “0.999…” isn’t actually a number (thus isn’t actually that sum). I agree that “0.999…” represents an infinite series summation (Σ 9/10^n), although there is no actual final sum to that series.

The set of numbers {.9, .99, .999, .9999, …} is nonempty and bounded above; therefore it has a least upper bound in the real numbers.

But 1.0 is bound upper and lower. A single number cannot be both completely bound and partially bound.

@wtf

Are you denying the completeness of the real numbers?

en.wikipedia.org/wiki/Completen … al_numbers

I’m saying that many of what we call “numbers” are not really numbers. They are not on the real number line. How could they be when they have no termination. We treat them as numbers because they are close enough for practical application (just as 3.25 used to be close enough to Pi). No truly infinite string of digits is an actual number on the real number line. Pi is a real number, but not ever represented in decimal form, just as is 1/3. Infinite strings of digits are “pseudo-numbers” (or “virtual numbers”) and do not represent real quantities.

@wtf

@James S Saint

Σ9/10^n = [0.999… :9R]

I’m not familiar with that notation and you haven’t defined it. What do the square brackets mean? What does the colon mean? What does the R mean?

I have defined it several times. The “R” merely indicates the “Ever Remaining Infinitesimal”. The colon, :, designates the separation between a proper decimal listing and the “ever remaining infinitesimal” region of the pseudo-number. There is nothing special about the brackets. They are just brackets for visual clarification (as often found in math).

Some virtual numbers have an ever remaining infinitesimal value that is different from the prior infinite string, such as :
Σ 1/2^n = [0.999…??? :75R]
That series has an ever remaining 75 infinitesimal. I put the “???” there because there is a list of varying numbers just before the 75 that depend upon to which decimal place you have carried the string.
0.875
0.9375
0.96875
0.984375
0.9921875
0.99609375
0.998046875
0.9990234375

And in this case, you know that this pseudo-number is less than [0.999… :9R] because those ??? digits are less than all 9s. And the properly placed digits leaves the ever remaining :75R to be always less than the :99R.

0.999
0.9999
0.99999
0.999999
0.9999999
0.99999999
0.999999999
0.9999999999

@wtf

@James S Saint

… so as to indicate that there is an ever remaining infinitesimal amount (and thus never truly summed).

Earlier you agreed that .999… = 1 is a theorem in the hyperreals. So what do you mean by infinitesimal in this context?

No. I didn’t. I said that it is only true for the hyperreals if it is true for the standard reals. And it isn’t true for the standard reals.

@wtf

And again, what do you mean by “truly?” So that I can understand you better, are you arguing a Platonist view that there is a mysterious, inaccessible realm of “true” real numbers “out there” that we have no access to? You keep using words like “truly” and “actual reals” when I have no idea what you mean and you haven’t taken the trouble to define your terms.

Before I slam you down once again, how about take a moment to examine exactly who is now insulting whom (again).

.
.

The word “true” means “in exact alignment with”. It is an ontology and logic term. In society, we say that something is true when it aligns with reality. A seaman says that his heading is true when it exactly aligns with a chosen compass setting. In logic and math we say that something is true when it is consistent with our axioms, declarations, definitions, and premises.

Mathematics has the concept of “real number” already defined (“In mathematics, a real number is a value that represents a quantity along a continuous line.”), so they are not free to just throw anything into that set (such as unbound, virtual, pseudo numbers). They are merely mathematicians, so they usually don’t know that, but it is “true to reality” that they are restricted if they want their math system to remain rationally usable.

[wtf1 (wtf)]

@James S Saint

No, no. Zeno’s is merely a high school calculus puzzle. Calculus is about the relation between two different infinitesimal variables.

False. You’re just exhibiting your lack of knowledge of the subject.

@James S Saint

In this case, both reducing at the exact same rate leaving a constant relation between them, speed.

False again. Exhibiting more lack of knowledge.

@James S Saint

But 1.0 is bound upper and lower. A single number cannot be both completely bound and partially bound.

Non sequitur. Gibberish.

@James S Saint

I’m saying that many of what we call “numbers” are not really numbers.

More of same.

@James S Saint

Before I slam you down once again

Were you raised in a barn? Do you have some kind of diagnosed personality defect?

[James_S_Saint]

Obviously you have a really hard time with losing a debate.

You might want to check that ego thing you have going.

[Carleas]

I’m not talking about time or speed, I’m talking about distance. There is 1 mile. Between 0 and 1 miles are an infinite series of distances 90% of the way from 0 to 1 mile. To go 1 mile, we must go an infinite number of these 90% segments. We don’t care how long or how fast we traverse these 90% segments, just that they all must be traversed. So we get an infinite sum .9 + .09 + .009 + .0009 + …

We know we’ve gone 1 mile, and that we’ve traversed an infinite number of these segments. But you seem to be saying that after traversing infinitely many segments, we still have some distance to cover. Moreover, you must be claiming that we can’t go 90% of the remaining distance (else, it would just be another 90% segment in the infinite sum); so there was a distance that could only be traversed all at once (i.e., a distance that was traversed without ever having completed 90% of that distance).

[James_S_Saint]

@Carleas

I’m not talking about time or speed, I’m talking about distance. There is 1 mile. Between 0 and 1 miles are an infinite series of distances 90% of the way from 0 to 1 mile. To go 1 mile, we must go an infinite number of these 90% segments. We don’t care how long or how fast we traverse these 90% segments, just that they all must be traversed. So we get an infinite sum .9 + .09 + .009 + .0009 + …

Zeno’s paradoxes were about motion and speed; tortoise and the hare, motion of the arrow, dichotomy…

@Carleas

We know we’ve gone 1 mile, and that we’ve traversed an infinite number of these segments. But you seem to be saying that after traversing infinitely many segments, we still have some distance to cover. Moreover, you must be claiming that we can’t go 90% of the remaining distance (else, it would just be another 90% segment in the infinite sum); so there was a distance that could only be traversed all at once (i.e., a distance that was traversed without ever having completed 90% of that distance).

If you are taking only an infinitesimal amount of time with each step, you can take an infinity of such steps in “real time”, finite time.

What you are probably not seeing (and yet talking about), is the fact that YOU are choosing to stop yourself at only 90% of a goal (or more properly 50% for Zeno) with each step. If you infinitely retard yourself, of course, you aren’t going to get anywhere. It is your own definition that forbids you from ever reaching 1.

If you are confused that there are points on the real number line that are only 90% of 90% of 90%… and you must somehow traverse such points, don’t be. Such points don’t actually exist as numbers on the real number line. 0.999… is merely a pseudo-number. It doesn’t actually exist. Nor does 0.555…, nor 0.111…

All you (or Zeno) are doing is confining yourself to seek out an infinity of steps that forbid you from getting to a chosen goal. Yes, there are an infinity of steps to be found. And taking only 90% is a certain way to make sure you find an infinity of steps that will not get you to the goal. Each of the individual points exists on the number line and there are an infinity of them … and actually many more than that (remember aleph-1). You could choose 50% for each step, 5%, or 1%. Each would yield an infinity of points that do not reach to the goal. All you are doing is restricting yourself to never getting to the goal. In effect, you are trying to count the infinitesimals between 0 and 1. But so what?

Zeno and Aristotle were talking about paradoxical motion. You seem to be talking about infinitesimal existence. In both cases, and infinity of infinitesimals forms a finite … paradox resolved.

[Carleas]

Agreed, Zeno’s paradox is inapposite and distracting, so let’s set it aside.

My point is, if we travel 1 mile and then ask, how many times during that trip did we go 90% of the remaining distance? We did so an infinite number of times. What I am saying, what wtf is saying, what the math world says, is that that infinite number of times is sufficient.

What you are saying is that it isn’t. After going 90% of the remaining distance an infinite number of times, you must posit a remaining distance yet to be traveled, and you must maintain that it is impossible to travel 90% of that distance without traveling all of it. Right?

[Meno2 (Meno_)]

I think the answer is that the sum of increments is a different kind of infinity than those of the sequences.
The bound infinity, which pertains to the integrated function, is modeled after linear thinking. But linear is really an arc of a circle, because there is no absolute
straight line. The model is conceptual.

Therefore the integral function of any two points on
the circle can be a spatial d(X) of any number of
spacial segments, even toward infinity, Bbut it is bound by a limit. That does not make it not infinitely divisible.

The other, the infinite sequence, has no limit at all, therefore, it is limitless, boundless. This infinity has
neither inside, or outside, therefore, it has only a
virtual ‘size’, or spatial determinant. Having none, it is beyond conception, modeling, linearly or otherwise.
It is a virtual, not conceptual infinity, because such
an infinity can not be conceived of.

How can the size of the universe be measured, when the only way is to measure to the limit of telescopic
view? What is beyond that, and beyond that, and
that? Is there even a beyond, if a limit can not be conceived? It becomes virtually impossible, within the limitations of language to define an endless infinity.
If it is said that such an infinity is endless, is it not
merely a hypothetical mathematical virtual conception, based on the idea that endless sequences can go on infinitely, and if they can not, can the
paradoxical halving of each subsequent element
defeat the idea of size even, if the sequence is carried on infinitely? (In Zeno’s case)

But if the size of even near an infinite calculation still retains some of it’s measurability, won’t it interfere with the number of times the sequence can still go in,
and still satisfy the criteria of both size and infinity?

Therefore, these are separate infinities, if at all it is what they are, and .
99999999999999999999999…cannot possibly equal 1.

[James_S_Saint]

@Carleas

Agreed, Zeno’s paradox is inapposite and distracting, so let’s set it aside.

My point is, if we travel 1 mile and then ask, how many times during that trip did we go 90% of the remaining distance? We did so an infinite number of times. What I am saying, what wtf is saying, what the math world says, is that that infinite number of times is sufficient.

What you are saying is that an endless number of times is sufficient. And what I am saying is that because it is endless, it never resolves and thus is insufficiently conclusive (thus cannot be said to be equal to anything). It is not a number.

@Carleas

What you are saying is that it isn’t. After going 90% of the remaining distance an infinite number of times, you must posit a remaining distance yet to be traveled, and you must maintain that it is impossible to travel 90% of that distance without traveling all of it. Right?

What I am saying is strictly what the term means, regardless of the consequences. The term “0.999…” has a meaning that involves an endless string of digits that never resolves to a value. YOU are saying that because we want this or that to work out later, we are going to accept that an endless string of digits is going to resolve to be equal to its apparent limit. You seem to have no argument as to why it would other than the desire that it does.

You are trying to assess the truth of logic based upon what you want to believe of the conclusion (argumentum ad consequentiam). Perhaps (and most probably) you are confused about the arguments concerning those other issues.

We can discuss the conundrum that you raise concerning the existence of infinitesimals, absolute zero, and how they play together after we settle whether the First Proof is logically valid.

@James S Saint

The logic of the First Proof is that the definition of an infinite continuation such as:
[list]wherein Σ is the accumulated sum at each stage
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
.
.
.[/list:u]
… forbids any change in the pattern.

Since the pattern directly indicates that ONLY 90% is ever to be taken, by definition, 100% can NEVER be taken.

The logic is not about the summation. It is about no change in pattern (exactly what the ellipsis is about).

The summation cannot be zero
because for it to become zero, the pattern would have to change yet it is forbidden to change.

That is not “begging the question”, but answering the question.

Is that logically valid or not?

@jerkey

there is no absolute straight line

As strange as that sounds, it is true.

[Carleas]

@James S Saint

The term “0.999…” has a meaning that involves an endless string of digits that never resolves to a value. YOU are saying that because we want this or that to work out later, we are going to accept that an endless string of digits is going to resolve to be equal to its apparent limit.

If you’ll look closely at my most recent post, you’ll notice that I did not mention .999…

Let’s start small: At some point in moving from 0 to 1, we go 90% of the remaining distance an infinite number of times, right? Surely we agree about that, and disagree about where between 0 and 1 that happens.

[James_S_Saint]

@James S Saint

What I am saying is strictly what the term means, regardless of the consequences. The term “0.999…” has a meaning that involves an endless string of digits that never resolves to a value.

You are trying to assess the truth of logic based upon what you want to believe of the conclusion (argumentum ad consequentiam). Perhaps (and most probably) you are confused about the arguments concerning those other issues.

We can discuss the conundrum that you raise concerning the existence of infinitesimals, absolute zero, and how they play together after we settle whether the First Proof is logically valid.

@James S Saint

The logic of the First Proof is that the definition of an infinite continuation such as:
[list]wherein Σ is the accumulated sum at each stage
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
.
.
.[/list:u]
… forbids any change in the pattern.

Since the pattern directly indicates that ONLY 90% is ever to be taken, by definition, 100% can NEVER be taken.

The logic is not about the summation. It is about no change in pattern (exactly what the ellipsis is about).

The summation cannot be zero
because for it to become zero, the pattern would have to change yet it is forbidden to change.

That is not “begging the question”, but answering the question.

Is that logically valid or not?

[Carleas]

The pattern is that the same thing happens at the end of each string of numbers. And you want that pattern not to change when we talk about an endless string. It’s not that it’s logically invalid, it’s that it’s incoherent. If there is no end to a string, then there is no fact of the matter about what happens at the end of that string.

As for what you’re describing as a “conundrum”, it is intended more as an intuition pump, to see what we’re really saying when we say that the sum of the infinite series equals/does not equal 1. It isn’t necessarily about infinitesimals or “absolute zero” (?), and certainly isn’t at this point:

@I

At some point in moving from 0 to 1, we go 90% of the remaining distance an infinite number of times, right?

[James_S_Saint]

@Carleas

The pattern is that the same thing happens at the end of each string of numbers. And you want that pattern not to change when we talk about an endless string. It’s not that it’s logically invalid, it’s that it’s incoherent. If there is no end to a string, then there is no fact of the matter about what happens at the end of that string.

I largely agree, but not entirely.
We have stated in the symbolism that the pattern never changes. I agree that we cannot say what happens “at the end”, because there is no end. The fact that there is no end for it is what prevents it from being the same as 1 (which obviously has an end). The same number cannot both have an end and not have an end (as per our agreement of non-contradiction).

Incoherency is logical invalidity/fallacy. If it is not logically invalid, it cannot also be incoherent.

So I ask again: Is the First Proof logically valid (regardless of the apparent consequancies)?

=======================

@Carleas

As for what you’re describing as a “conundrum”, it is intended more as an intuition pump, to see what we’re really saying when we say that the sum of the infinite series equals/does not equal 1. It isn’t necessarily about infinitesimals or “absolute zero” (?), and certainly isn’t at this point:

@I

At some point in moving from 0 to 1, we go 90% of the remaining distance an infinite number of times, right?

How we get from 90% of 90% of 90%… to 100% is another issue. … and rather interesting in its own right. A good topic for the next thread.

But it is invalid to claim that "Columbus’ reasoning must be wrong else everyone would fall off of the Earth".

[Carleas]

@James S Saint

How we get from 90% of 90% of 90%… to 100% is another issue.

What? That is exactly the same issue as “get[ting] from” .999… to 1.

@James S Saint

Incoherency is logical invalidity/fallacy.

OK. It’s invalid.

@James S Saint

The same number cannot both have an end and not have an end (as per our agreement of non-contradiction).

I don’t grant this. There is no contradiction in having multiple ways to express the same number. 2/2 = c^0 = 0! = 1 are all expressions of the same value. Different number systems have different repeating decimals. In base 3, 1/3 is .1, so it’s expressible with or without the repetition.

My assertion here is that .999… and 1 are both decimal expansions of the same number.

Thinking about bases, I have another intuition pump for us. True or false:

.444… [base 5] = .999… [base 10]

I say yes. But maybe we should start simpler. True or false:

10 [base 5] = 5 [base 10]

I say yes again.

[James_S_Saint]

@Carleas

@James S Saint

How we get from 90% of 90% of 90%… to 100% is another issue.

What? That is exactly the same issue as “get[ting] from” .999… to 1.

No. It certainly is not.

[list][b]The debate is whether 1 IS 0.999… in value

It is NOT how someone traverses 0.999… to get to 1.[/b][/list:u]

That is an ontology issue, not an equality issue.

@Carleas

@James S Saint

Incoherency is logical invalidity/fallacy.

OK. It’s invalid.

Then point out the invalid statement(s) … the inconsistency.

You are seeing it as incoherent because you see that the conclusion doesn’t fit into your delusions of how the universe works. It is not coherent with your belief system. Of course nothing could ever possibly be wrong with your belief system.

@Carleas

@James S Saint

The same number cannot both have an end and not have an end (as per our agreement of non-contradiction).

I don’t grant this. There is no contradiction in having multiple ways to express the same number. 2/2 = c^0 = 0! = 1 are all expressions of the same value.

We are not talking about how it is expressed. We are talking about WHETHER THEY ARE EQUAL VALUES!!
But one of the expressions is of an endless sum, leaving the represented number boundless and unresolved.

@Carleas

My assertion here is that .999… and 1 are both decimal expansions of the same number.

And yet has been proven wrong several ways.

Point out what statement in this proof is invalid (stop dodging and strawmanning):

@James S Saint

What I am saying is strictly what the term means, regardless of the consequences. The term “0.999…” has a meaning that involves an endless string of digits that never resolves to a value.

You are trying to assess the truth of logic based upon what you want to believe of the conclusion (argumentum ad consequentiam). Perhaps (and most probably) you are confused about the arguments concerning those other issues.

We can discuss the conundrum that you raise concerning the existence of infinitesimals, absolute zero, and how they play together after we settle whether the First Proof is logically valid.

@James S Saint

The logic of the First Proof is that the definition of an infinite continuation such as:
[list]wherein Σ is the accumulated sum at each stage
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
.
.
.[/list:u]
… forbids any change in the pattern.

Since the pattern directly indicates that ONLY 90% is ever to be taken, by definition, 100% can NEVER be taken.

The logic is not about the summation. It is about no change in pattern (exactly what the ellipsis is about).

The summation cannot be zero
because for it to become zero, the pattern would have to change yet it is forbidden to change.

That is not “begging the question”, but answering the question.

[Carleas]

@James S Saint

Then point out the invalid statement(s) … the inconsistency.

When you suggest that something happens at the end of an endless string. That’s inconsistent.

[James_S_Saint]

@Carleas

@James S Saint

Then point out the invalid statement(s) … the inconsistency.

When you suggest that something happens at the end of an endless string. That’s inconsistent.

So, never mind that it is YOU who suggests that something out of the ordinary (an alteration of the pattern) occurs.

YOU are the one saying that something magical happens “at infinity”.

I am saying that nothing changes AT ALL. The exact same pattern continues forever, just as the ellipsis indicates.

So what were you lying about?

I asked you a simple question:
“…point out the invalid statement(s) … the inconsistency.”

And your only response is to accuse me of what YOU are guilty of doing … suggesting that the pattern magically changes “at infinity” … whatever the hell “at infinity” is supposed to mean???

YOU are suggesting that there is this magical “at infinity” where-at the pattern is to change from taking only 90% to taking 100%. That presumption on YOUR part is inconsistent, contradictory, and invalid. Nothing suggests any change what so ever in the pattern of taking only 90%.

I have to accept that you have no rational response to the First Proof of the opposition to your opinion.
You have chosen (no big surprise) to be merely a “Nay Sayer” for the duration of this debate.

=====================

So now the next question:
The Second Proof:

@James S Saint

2)
Second Proof that the two “numbers” are not equal, the “set difference definition proof”.
The definition of the difference of the sets is:
1.0 - 0.9 = 0.1
1.0 - 0.99 = 0.01
1.0 - 0.999 = 0.001
1.0 - 0.9999 = 0.0001
.
.
1.0 – 0.999… = 0.000…1
That is an infinite set of non-zero items.
For 1.0 to equal 0.999…, there must be a zero difference. QED

So of what guilt are you going to do and accuse of me concerning this issue?

[Uccisore]

@James S Saint

[

If you are confused that there are points on the real number line that are only 90% of 90% of 90%… and you must somehow traverse such points, don’t be. Such points don’t actually exist as numbers on the real number line. 0.999… is merely a pseudo-number. It doesn’t actually exist. Nor does 0.555…, nor 0.111…

Every decimal amount that you think 'actually exists is expressed as a repeating decimal in some other base. Good old 0.5 is 0.111… in ternary (base 3). Does that mean .5 doesn’t exist, or that mathematicians are wrong that .5 is expressed as .111… in ternary? In ternary, 0.222… = 1 just like .999… = 1 in our decimal system.

Likewise, 1/3rd, this mysterious .3333… that you would say doesn’t exist, is simply 0.1 in ternary. Same value- the value you get if you split 1 into three equal portions, or, the value such that when you multiply it by three, you get one. If 0.3333 is a ‘pseudo number’, then 0.1 must be as well (in ternary) and yet the only reason you call it one is because of how it repeats.

“.999…” expresses a value equal to 1. It doesn’t express ‘nothing at all’, it doesn’t express something imprecise, there is no infinite series of nines that we have to account for with philosophy. Because we happen to use a base ten system, certain fractional amounts look odd when written out as a decimal. IF we used a base-something-else system, then those fractions would look fine, and certain other ones would look odd when written out. This is all a bunch of fuss about nothing.

1/3rd looks funny as a decimal. When you multiply that decimal by three, that looks funny too, but if you’ve made it through grade 4 you know the result must be 1. The ‘decimal’ expression of 1/3rd doesn’t look funny at all in ternary, though.

Decimal: 1/3 = .333… .333…x 3 = .9999 = 1
Ternary: 1/10 = .1 .1 x 10 = 1

These both mean the exact same thing.

Decimal: 1/2 = .5 .5x2 = 1
Ternary: 1/2= .111… .111…x2 = .222… = 1

These both mean the exact same thing.

[James_S_Saint]

@Uccisore

@James S Saint

[

If you are confused that there are points on the real number line that are only 90% of 90% of 90%… and you must somehow traverse such points, don’t be. Such points don’t actually exist as numbers on the real number line. 0.999… is merely a pseudo-number. It doesn’t actually exist. Nor does 0.555…, nor 0.111…

Every decimal amount that you think 'actually exists is expressed as a repeating decimal in some other base. Good old 0.5 is 0.111… in ternary (base 3). Does that mean .5 doesn’t exist, or that mathematicians are wrong that .5 is expressed as .111… in ternary? In ternary, 0.222… = 1 just like .999… = 1 in our decimal system.

Ucc, no offense, but it is seriously kind of dumb to suggest that "if I said those same words in Chinese, they would mean something true" as your argument. We are speaking of and in one particular language and “base”. It really doesn’t matter what some Martian might read it it as. It doesn’t matter what is true IF, IF, IF it was stated in octal.

Your argument is entirely “begging the question”.

[Uccisore]

@James S Saint

Ucc, no offense, but it is seriously kind of dumb to suggest that "if I said those same words in Chinese, they would mean something true" as your argument. We are speaking of and in one particular language and “base”. It really doesn’t matter what some Martian might read it it as. It doesn’t matter what is true IF, IF, IF it was stated in octal.

You should have paid more attention. Now I have to defeat your position again with the same argument, but more words. I’m not going to do it a third time.

I’m not saying “If I said those same words in Chinese, they would mean something true”. Even in english, even in base ten, your argument is already shit if you simply convert .333… to 1/3rd. That should have been the end of it. But I’m reminding you that even if you are absolutely hung up on decimals (because they are the only way to make it appear as though you have a point), there are ways to represent these same values that shows the flaw in your reasoning.

The symbol ‘1’ in ternary and decimal indicate the same amount- one.

.1 in ternary means the exact same thing as .333… means in decimal. They both indicate precisely the amount: the size of one of the parts of a whole divided into three equal parts. 1/3rd (1/10th in ternary). If one of them ‘isn’t a real number’, than the other isn’t either, because they are equal.

Similarly, .5 in decimal is .111… in ternary. Again, they both mean precisely the same amount: the size of one of the parts of a whole divided into two equal parts. If .111… isn’t a real value in ternary, than .5 isn’t in decimal, because they are the same value.

Since they are the same value, when you double them (or whatever), you also get equivalent values. .111… doubled in ternary is .222… This means that if you double .5 in decimal, the result has the same value as .222… in ternary. .5 doubled in decimal is 1. That means .222… in ternary is 1.

Likewise, when you multiply .333… by three you get .999…

When you multiply. .1 by three in ternary, you get 1.

Since .1 in ternary is .333…, and since ‘1’ in both ternary and decimal represent the same amount, .333… multipled by three in decimal must equal 1. And so .999… must equal 1. And as it turns out, it does.

The point here is that you are trying to argue that .999… (or .333… or any other such number) has mysterious properties that make it different from other numbers, and is not a real number at all. The problem of course is that “.333…” is just a representation of an amount- and there’s a shitload of other ways to represent that same amount. 1/3rd is one. .1 is another (in ternary). When we use these other, simpler representations, we easily see that the value represented by .333… tripled is 1. Of course it is the values these symbols represent that makes “.999… equals one” true.

Simply put, “.999… equals one” is excedingly easy to demonstrate using fractions, using ternary, and probably using other ways of expressing these values. Unless you’re going to prove that .333… is not equal to 1/3rd, and not equal to .1 in ternary, you have no argument.

.333… is not mysterious. It is not a non-number, or an imprecise value. It is the value indicated by 1/3rd. It is the value indicated by “one third”. It is the value indicated by .1 in ternary. It is a value any child can grasp. You’re basing your entire argument out of presenting a value in the least-clear way we have available, and then trying to tease implications out of the lack of clarity.

[James_S_Saint]

@Uccisore

@James S Saint

Ucc, no offense, but it is seriously kind of dumb to suggest that "if I said those same words in Chinese, they would mean something true" as your argument. We are speaking of and in one particular language and “base”. It really doesn’t matter what some Martian might read it it as. It doesn’t matter what is true IF, IF, IF it was stated in octal.

You should have paid more attention. Now I have to defeat your position again with the same argument, but more words. I’m not going to do it a third time.

You haven’t done it once yet. You are failing to see your logic fallacies (from not reading the thread).

@Uccisore

I’m not saying “If I said those same words in Chinese, they would mean something true”. Even in english, even in base ten, your argument is already shit if you simply convert .333… to 1/3rd.

Which you
can
not
do
… as discussed much earlier in this same thread.

@Uccisore

.1 in ternary means the exact same thing as .333… means in decimal.

No. It doesn’t. That is your logic fallacy called “begging the question” (petitio principii). What doesn’t work for 0.999… also doesn’t work (for the same reason) for 0.333… and 0.111… and anything else ending with “…”.

What you are presuming is that the “…” ellipsis represents an infinite series that magically terminates “at infinity”. There is no “at infinity”. The series NEVER ENDS. And thus it never resolves to be equal to anything.

That is what this First Proof is saying (and has been for several pages):

@James S Saint

What I am saying is strictly what the term means, regardless of the consequences. The term “0.999…” has a meaning that involves an endless string of digits that never resolves to a value.

You are trying to assess the truth of logic based upon what you want to believe of the conclusion (argumentum ad consequentiam). Perhaps (and most probably) you are confused about the arguments concerning those other issues.

We can discuss the conundrum that you raise concerning the existence of infinitesimals, absolute zero, and how they play together after we settle whether the First Proof is logically valid.

@James S Saint

The logic of the First Proof is that the definition of an infinite continuation such as:
[list]wherein Σ is the accumulated sum at each stage
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
.
.
.[/list:u]
… forbids any change in the pattern.

Since the pattern directly indicates that ONLY 90% is ever to be taken, by definition, 100% can NEVER be taken.

The logic is not about the summation. It is about no change in pattern (exactly what the ellipsis is about).

The summation cannot be zero
because for it to become zero, the pattern would have to change yet it is forbidden to change.

That is not “begging the question”, but answering the question.

Rather than trying to overlay your own reasoning. How about find something wrong with that First Proof (besides the fact that you don’t like the conclusion).