## View Poll Results: Is 0.999... exactly equal to one?

Voters
86. You may not vote on this poll
• Yes, they are equal.

52 60.47%
• No, they are not equal.

34 39.53%

# Thread: Is 0.999... equal to one?

1. ## Re: Is 0.999... equal to one?

I'm not attracted by finitism (and not just because it seems to need invective to be expressed).

I don't think "exists" can be applied to mathematical concepts, even the natural numbers. I'd say that they are answers to questions - in the case of the natural numbers, "how many". More complex questions require different concepts, and I see the jump to infinity, perhaps unbridgeable in terms of existence, as simply the result of asking more demanding questions.

I see mathematics as having the form: "If I say X then I am compelled to say Y" (X being only limited by Y not containing not-X, not by the plausibility of X).

As for Dedekind cuts, I can see that finitists will reject the concept, whereas I think it answers a difficult question about defining arithmetic for what the finitists call magnitudes, and therefore about the centrally important ordered complete field we call the "reals".

If I allow myself to talk about infinite sets of rationals then there are two exemplary cases:

for zero, the lower set is all the negative rationals and the upper set is zero itself and all the positive rationals.

for root two, the lower set is all the rationals whose square is less than two and the upper set is all the rationals whose square is greater than two.

Then it's actually quite easy (always accepting that we can talk about infinite sets) to define arithmetic with lower sets, the operations yielding the appropriate lower set, and therefore a cut.

Sadly, I know that this post, which I have tried to make as uncontentious as possible, will not be found acceptable.

2. ## Re: Is 0.999... equal to one?

Originally Posted by agingjb
I'm not attracted by finitism (and not just because it seems to need invective to be expressed).
What makes you think it needs invective to be expressed? "Infinitism" does a pretty solid job of denouncing itself.

Originally Posted by agingjb
I don't think "exists" can be applied to mathematical concepts, even the natural numbers.
Of course it can. It is central to mathematical concepts that are well defined. The natural numbers have always existed because they are well defined.

Originally Posted by agingjb
I'd say that they are answers to questions - in the case of the natural numbers, "how many".
That's extremely vague. The natural numbers exist as a consequence of measure in terms of whole units.

Originally Posted by agingjb
More complex questions require different concepts, and I see the jump to infinity, perhaps unbridgeable in terms of existence, as simply the result of asking more demanding questions.
Well, you are clearly confused. Mathematics is not about asking questions. It's the study of measurement and the properties of numbers that arise as a result thereof. Questions are asked in every field of knowledge but have their roots in philosophy, which is the source of all knowledge.

Originally Posted by agingjb
I see mathematics as having the form: "If I say X then I am compelled to say Y" (X being only limited by Y not containing not-X, not by the plausibility of X).
You are thinking of logic which is one of the main tools of mathematics, but by no means that which defines it.

Originally Posted by agingjb
As for Dedekind cuts, I can see that finitists will reject the concept, whereas I think it answers a difficult question about defining arithmetic for what the finitists call magnitudes, and therefore about the centrally important ordered complete field we call the "reals".
Um, no. What you think of as an ordered field is clearly ill-defined nonsense because you don't understand what is a magnitude. There is no question about defining arithmetic for magnitudes - arithmetic is ONLY defined for numbers. Magnitudes are not numbers unless they have been measured.

Originally Posted by agingjb
If I allow myself to talk about infinite sets of rationals then there are two exemplary cases:

for zero, the lower set is all the negative rationals and the upper set is zero itself and all the positive rationals.
You are such a pompous ass. "if YOU allow yourself"?! Tsk, tsk. You obviously do not understand what is a Dedekind cut. May I suggest some serious study of my previous comments in that regard?

Originally Posted by agingjb
Then it's actually quite easy (always accepting that we can talk about infinite sets) to define arithmetic with lower sets, the operations yielding the appropriate lower set, and therefore a cut.
You can talk about many things, but doing so does not mean you know what you are talking about.

Originally Posted by agingjb
Sadly, I know that this post, which I have tried to make as uncontentious as possible, will not be found acceptable.
If you call this uncontentious, I would hate to see one of your contentious posts. May I remind you once again Aging JB, mathematics is not about opinion or like or dislike, it is about cold, hard facts and concepts that are well defined.

One can only ever be certain that concepts and ideas exist if and only if, the same are well defined.

The rest is simply idle horseshit! (Oh, please do pardon the invective!).

3. ## Re: Is 0.999... equal to one?

Originally Posted by john_gabriel
What makes you think it needs invective to be expressed?
If finitism doesn't need invective to be expressed, why do you keep saying things like

You are such a pompous ass.
and

The rest is simply idle horseshit! (Oh, please do pardon the invective!).
What is the need for the invective?

4. ## Re: Is 0.999... equal to one?

Originally Posted by Colin
If finitism doesn't need invective to be expressed, why do you keep saying things like

and

What is the need for the invective?
Probably because I do not consider it to be invective. Would you like to know what I consider to be invective?

I didn't think so!

Say Colin, what kind of tone does the following phrase carry?

If I allow myself to talk about infinite sets... - Aging JB

Pompous ass?

It is extremely arrogant and condescending. You're so naive that you don't realize it. Tsk, Tsk.

Do you think anyone can prevent him from talking about infinite sets or anything else?
Do you think that perhaps he thinks it's beneath him and that he might be mocking everyone else?
Do you think that perhaps his nose is turned upwards so much that he can't see everyone else around him?

Do you need more clues Colin? You have not answered any of the previous refutations of your wrong arguments. Not that it matters, but it's so unlike you not to.

5. ## Re: Is 0.999... equal to one?

Originally Posted by agingjb
Sadly, I know that this post, which I have tried to make as uncontentious as possible, will not be found acceptable.
It appears that you have made an accurate mathematical prediction.

6. ## Re: Is 0.999... equal to one?

Originally Posted by Neverfly
It appears that you have made an accurate mathematical prediction.
Well, you must be able to see something I've missed. None of Aging JB's comments contain any math.

7. ## Re: Is 0.999... equal to one?

Originally Posted by john_gabriel
pi as an approximation can be used to describe the length of a line, the size of an angle, the volume of a cylinder or the amount of paint in a can. pi is not a number. However, an approximation of the magnitude pi is a number.

In the statement:

The circumference of a circle of 1 meter diameter is pi meters.

Are the expressions "1 meter" and "pi meters" different in character? Is either of these expressions any more or less accurate than the other? Is either expression an approximation?

Or consider the following 2 statements

Each angle of an equilateral triangle is 60 arc degrees.

Each angle of an equilateral triangle is pi/3 radians.

Are the expressions "60 arc degrees" and "pi/3 radians" different in character? Is either of these expressions any more or less accurate than the other? Is either expression an approximation?

8. ## Re: Is 0.999... equal to one?

Originally Posted by Colin
In the statement:

The circumference of a circle of 1 meter diameter is pi meters.

Are the expressions "1 meter" and "pi meters" different in character? Is either of these expressions any more or less accurate than the other? Is either expression an approximation?

Or consider the following 2 statements

Each angle of an equilateral triangle is 60 arc degrees.

Each angle of an equilateral triangle is pi/3 radians.

Are the expressions "60 arc degrees" and "pi/3 radians" different in character? Is either of these expressions any more or less accurate than the other? Is either expression an approximation?
~ Yes Colin if the diameter is 1. meter the circumference must be 3.14... meters.

If you DO NOT make a approximation of that sum it continues to infinity... undefined.

In ALL cases of working with pi we make that approximation. Are you confused by pi ?

~ Yes Colin the deg., of a equilateral triangle must be of 60 deg..each.

~ I can not answer what I do not comprehend.. pi/3 radians is not a term I would use.

I meter is a well defined sum, while pi meters is not.

9. ## Re: Is 0.999... equal to one?

Originally Posted by Colin
In the statement:

The circumference of a circle of 1 meter diameter is pi meters.

Are the expressions "1 meter" and "pi meters" different in character? Is either of these expressions any more or less accurate than the other? Is either expression an approximation?
Yes. 1 meter is well defined. pi meters is not.

Originally Posted by Colin
Or consider the following 2 statements

Each angle of an equilateral triangle is 60 arc degrees.

Each angle of an equilateral triangle is pi/3 radians.

Are the expressions "60 arc degrees" and "pi/3 radians" different in character? Is either of these expressions any more or less accurate than the other? Is either expression an approximation?
There is no such thing as "60 arc degrees". If you mean 60 degrees and pi/3 radians, then YES, they are different in character - the one is a number and the other is not.

By the way, the number of degrees chosen in a circle is arbitrary. I could have chosen 30pi degrees for each circle, in which case, an equilateral triangle would have 5pi degree angles.

Perhaps in some other solar system, they use the convention of 30pi degrees in each circle. Ha, ha.
So tempted to use a nasty invective here, but I will refrain!

I am still waiting on for your response to my proof that Dedekind cuts do not define real numbers. Or is that simply something you can't fathom, so you've chosen to ignore it?

"Real numbers" as you imagine them, do not exist. Neither Dedekind cuts nor Cauchy sequences define a "real" number.

is well defined if and only if, m is an integer on completion of the limit.

For if m is not an integer, then there is no sense in calling

definitions of different "real" numbers k, l, p and q because m is the same for them all, that is, . And pray tell, what sort of cut is ? Ha, ha.

I love it when ignorant modern mathematicians are so routed using their very own ill-defined concepts and ideas. Too funny!

So, let's see what happens to the rest of your simian mathematics when one of your fundamental notions fails (the fact that real numbers do not exist):

1. All the theory derived based on the ill-defined concept of real number is suspect.
2. The transfer principle is bollocks.
3. Infinitesimals are a dream because they are a subset of the "real" interval (0,1).
4. Hyperreals, surreals, incredibly stupid "reals" and all their simian relatives also do not exist.

Tsk, tsk. Colin. Tsk, tsk. Primate mathematicians are unbelievably stupid, aren't they?

Caveman1917 was blurting off his ignorance about cardinality and other bullshit. Turns out the moron did not even understand the basic concepts in his theory. Hilarious!

What truly astounds me is how these modern academic idiots have not yet capitulated in the face of mountains of evidence that debunks all their wrong ideas!

Test:

Answer follows at end of comment.

Choose ONLY the one most correct answer:

In the light of all this evidence, the theory of real analysis:

a) Must be sound and reliable.
b) Has some quirks but works most of the time.
c) Is nonsense because it is based on ill-defined concepts.
d) Is rot because John Gabriel claims it is.
e) (c) and (d).

My Asian students love these type of questions where they have to determine the most correct. They find the simple ones too boring. Well, no wonder so many of them score 2400 on SAT II.

Sigh, wouldn't it be nice if American students could someday be the same? Dream on...

Choice (e) is the correct answer.

10. ## Re: Is 0.999... equal to one?

Originally Posted by caveman1917
Yet you are not . Forms of a.bc... are quite obviously strings of symbols (to be precise, they are elements of a formal language) while a real number is an element of a model over that formal language. Models aren't unique, that's why we can't identify the model with the formal language. Especially in our case since the hyperreal numbers are an alternative model to the real numbers over the same formal language.

A model (or a structure) over a formal language is defined by an interpretation function, that's what are.
Yes, we use strings of characters to write and communicate, but it is valid to say red is a color, rather than have to say red is a string of characters that we interpret as the name of a color. Just as it is valid to say 0.333... is a number, whereas "0.333..." is a string of characters. In my posts I thought I was careful to distinguish between the two (or three) cases.

I don't think the next step is valid. Hyperreals addition is component-wise. In other words, the sum of the hyperreals (3, 3, 3, ...) and (.1, .1, .1, ...) is (3.1, 3.1, 3.1, ...). Another way of saying it, is, as hyperreals, 3 + .1 = 3.1. It's what you'd expect as images of reals in the hyperreals.

So the RHS of both are equal to eachother, and thus Pi is either equal to both or equal to neither.
I disagree.

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