The question is predicated on a faulty assumption.

Number lines do not exist (or, at least, cannot be described).

Definitions

All definitions are of the form: X is not(Everything Else)

The real numbers as intrinsic values are a phantom; an illusion.

When we describe numbers we actually describe the relationships of numbers.

'1' is understood by its relationships with '2', '3', '-4.8776', 'apples', ...

When we talk about X we are actually talking about X's relationships.

Infinitesimals

While it is worth considering what the smallest possible relationship is...

The difference between 0.999 recurring and 1 is (in part) their relationship(s) to each other.

The relationship between 0.999... and 1 defines a difference between them.

Indistinguishable

A relationship and a difference are, for all practical purposes, the same thing.

The presence of a relationship between two perceptions demonstrates a difference between those perceptions. We can distinguish between them.

0.999... is different from 1 because we perceive a difference.

The nature of difference

'Difference' is defined as not(not Difference).

0.999... is different to 1 - but we can only understand that difference by comparing it to other differences.

Sorry to be weird at you

I know this isn't the sort of answer you were looking for - but the question is only troublesome because of the mistaken belief that number lines exist.

It is flat impossible to objectively describe number lines (or anything else).

All descriptions are of relationships. We can describe the relationships of numbers. And we can describe relationships in comparison to other relationships.

This is the structure of all knowledge. This is the mechanism of understanding.

The real number line, as an objective entity has never been described; can never be described.

We can, and do, describe the relationships of numbers. This dense network of relationships is our concept of what numbers are.

I did not participate in the previous thread, so I don't know if this was discussed. 0.99999... is equal to the summation from n = 1 to infinity of 9/(10^n). Sorry, I don't know how to write that here in math symbols. It's been a long time since I did that kind of math, so we'll have to trust Chat GPT. It says it equals 1.

You can prove it pretty succinctly

.9999... = x
9.9999... = 10x
10x-x = 9.999... - .999...
9x = 9
x = 1

I take it that you took that info from my post.
But you made it the other way around.

Briefly, the position appears to be that in the (classical) real number line, 0.999... is the largest real number which is less than 1; Cantor's Diagonal Argument certainly seems to support this interpretation, — alan1000

0.999... is equal to 1 here, not lesser than 1.

Abraham Robinson's definition of h revolutionised mathematics in the 1960's. Briefly, he defined the infinitesimal as a number which, for all values of a, is <a and >-a. Thus the infinitesimal may have a range of values, including 0. Within THIS number line, it appears to be undeniable that 0.999... meets the limit of 1, and thus 0.999...=1. — alan1000

Here, 0.999... can indeed be less than 1, because 0.999... is ambiguous.

I'm sorry, but none of the replies so far seem to evidence any familiarity with number theory or basic set theory... in fact, I can't really identify any rational position to which I can respond... is there anybody out there who is familiar with this area of mathematical philosophy? I mean, when a respondent asserts (without supporting arguments) that number lines don't exist, how does one begin to frame a reply?

Number lines do not exist (or, at least, cannot be described).

Wow, I did not see that coming. More than 2500 years of mathematical development flushed down the toilet in a few seconds!

The real numbers as intrinsic values are a phantom; an illusion.

What are your supporting arguments? Give me some help here, I'm trying to understand your position.

All definitions are of the form: X is not(Everything Else)

The assertion that X has no identifying properties in its own right is certainly a courageous approach. To my knowledge, nobody in the previous history of mathematics, from Euclid to Penrose, has ever adopted such a definition. By this method, how would you define "prime number", for example?

When we describe numbers we actually describe the relationships of numbers.

"Relationships of numbers" is a defining property of the relational number line (the line of negative and positive integers). But you deny the existence of number lines. Can you develop this point?

The example provided by Flannel Jesus conforms with the parameters of the hyperreal number line, as set out in my original post. Of course, it does not call on any infinite or infinitesimal values, but if he wishes to argue therefrom that it is consistent with the logical properties of the real number line, he will need to provide clarifying arguments.

TC: I'm sorry, but "We'll have to trust Chat GPT" is not a philosophical argument. I'm not looking for someone to tell me the "right" answer; I am posing a question in the (apparently vain) hope that someone out there actually understands mathematical philosophy.

Actually, I was too rash in stating that Flannel Jesus' reply did not call on any infinitesimal values, ie, did not depend upon h. Of course, his conclusion does depend upon defining the infinitesimal as h, and thus falls within the scope of the hyperreal number line, within which context 0.999...=1, as I asserted in my original post.

I'm sorry, but none of the replies so far seem to evidence any familiarity with number theory or basic set theory... — alan1000

I think it is you who doesn't have any familiarity with high school math:

Briefly, the position appears to be that in the (classical) real number line, 0.999... is the largest real number which is less than 1 — alan1000

For the real numbers, 0.999... is exactly equal to 1.

I'm sorry, but none of the replies so far seem to evidence any familiarity with number theory or basic set theory... — alan1000

@flannel jesus gave a clear and obviously correct answer using simple arithmetic operations on real numbers. What does number theory have to do with it?

Abraham Robinson's definition of h revolutionised mathematics in the 1960's. — alan1000

This is an exaggeration. There are probably universities around where this is taught regularly, but it has not caught on to any significant degree in general. A colleague of mine who taught at the U of Colorado told me they made an attempt to start a course in the subject, but it flopped. I don't see any course in their curriculum now that focuses on non-standard analysis. But there are courses in foundations where it may crop up.

So, rather than drift off into systems that depart from the standard material on the real numbers, its best to stick with the widely accepted ideas. Just my opinion.

Even then, the claim that 0.999...<1 in the hyperreals isn't exactly correct either.

we'll have to trust Chat GPT. — T Clark

Even though it is right, its authority cannot be assumed. It confabulates.

Even though it is right, its authority cannot be assumed. It confabulates. — Banno

Nicely put. I've been searching for the right word and that's it.

I've used that word twice already this morning... Seems to be the word of the day. Perhaps age has given me a better understanding of its meaning.

@Jamal, I'm having visions of the forums being overtaken by the self-replicating grey goo of misnamed threads concerning 0.9999....

...if the moderators are friendly — alan1000

Let's hope they are not.

I'm having visions of the forums being overtaken by the self-replicating grey goo of misnamed threads concerning 0.9999... — Banno

Folks are abducted by the philosophy of mathematics during summertime. The effect of being burned down by the sun...

By "a peculiar turn of events", this mathematics (calculus) has been taken up extensively by physicists (and other scientists), showing that it indeed works to a tee, thereby indirectly giving these mathematical methods empirical support. ;) Cool eh?

You have it backwards.

$1 - 0.999… = x$

With the standard reals this equation can only be:

$1 - 0.999… = 0.000… = 0$

There is no standard real number greater than 0 that can satisfy the equation.

But with the hyperreals this can be:

$1 - 0.\underbrace{999…}_H = {1\over{10^H}}$

Where H is an infinite hyperinteger.

What is worth discussing is why the layman’s mathematical intuitions favour nonstandard analysis, and why standard analysis is standard.

What is worth discussing is why the layman’s mathematical intuitions favour nonstandard analysis — Michael

The same intuition that stands under Zeno's paradox.

why standard analysis is standard — Michael

Calculus and its wide applications, I would imagine.

Wow, I did not see that coming. More than 2500 years of mathematical development flushed down the toilet in a few seconds! — alan1000

Yup. Sort of. Descartes, Gödel, Alfred North Whitehead and The Foundational Crisis In Mathematics (among many others) have covered much of this territory before. It gets ignored because it is inconvenient - but this isn't entirely new.

Evidence

Try creating a non-circular definition.

Different approaches

Gödel's incompleteness theorems talk about the limitations of a system referring to itself.

Descartes observes that nothing can be proven outside one's own existence.

Alfred North Whitehead formalised process philosophy.

The foundational Crisis in Mathematics is a number of different people pointing out that axiomatic mathematics cannot establish a firm foundation from which to proceed.

General Relativity demonstrates that the assumptions of Newtonian Mechanics do not apply to the universe.

Context matters: the meaning of a thing depends (entirely) on the context.

Subjective experience exists.

Count Timothy von Icarus provides several relevant quotes in response to NOS4A2

Precedent

This really isn't out of the blue.

Mathematicians have tried really hard to establish a set of definite, unambiguous axioms. It can't be done.

Every statement within a (closed) system is one part of the system describing other parts.

All definitions are necessarily circular.

More pertinently: every definition is by reference to other things.

Bold, Underline & Italics

We can describe the relationships of X. We cannot describe X.

There is nothing complicated here. Whatever X is - we cannot access it. We cannot experience X. We cannot describe X.

In this respect I am simply reiterating an observation that is over two millennia old.

"Relationships of numbers" is a defining property of the relational number line (the line of negative and positive integers). But you deny the existence of number lines. Can you develop this point? — alan1000

We cannot describe X. We can describe the relationships of X.

The majority of the time this distinction doesn't matter. When sitting at the dining table it would be redundantly pedantic to note that we are experiencing the dining table's relationships rather than the dining table itself.

However, in pure mathematics, philosophy and metalanguage discussions the distinction becomes crucial. Such as when we are discussing whether 0.999... = 1.

Relative vs Objective: Change vs Static

The Law of Identity states that objects (like the number line) are static, unchanging.

A defining characteristic of relationships is that they change.

As you change, grow and learn, your relationships with concepts changes. Your understanding of numbers now is significantly different than when you were first being taught to add and subtract.

Each person has (slightly) different ideas of what numbers are, and the significance of them.

There are, of course, similarities. Common experiences create similar networks of relationships. An ordered series of numbers (1<2<3<4<5<...) is a near universal experience. It is easy to confuse many apparently similar subjective experiences for objective truth.

Your understanding of the number line is dynamic. Your sense of knowledge and meaning changes.

The concept of "number line" that you posses is constantly evolving, developing, changing.

In closing

That all concepts (such as number lines) are dynamic is in direct opposition to The Law of Identity.

It is a big step to swallow in one go.

However, the individual components are simple enough observations:

• Meaning is defined by context.
• Understanding is subjective.
• The universe changes.

These aren't shocking, groundbreaking revelations.

It just so happens that a static number line with fixed (true) rules is a direct contradiction of these observations.

It seems to me that geometry/space is what has presente a foundation for all mathematics. If numbers and sets of numbers alone was the concern, could not mathematics procede perfectly fine without paradox or bothering with Godel? Looking back on itself wouldn't even be necessary. Once you start looking for ratios of angles ect. it leads to area wherein an infinite space can be enveloped by finite territory. If only manbdid not measure space, maybe numbers would never have become a problem

"The set of all sets that do not contain themselves". Obviously this top set could not self reference. I would say the same of Godel

My 2 cents

As with Zeno's paradoxes where we see space dissolve into nothing (or parmendian pure being), numbers must have a basic unity that holds them from infinite divisione. If we have 1, then we have 2 halfs, which each is one, so 1 is two. This can go on forever- as with divisione of a line. Numbers are synthetic (Kant)and nah platonic (Plato). A number is not a set. 7 is not the set of 7 ones. Sets are applied by us TO numbers which WE can choose how to group

I would say that whether 0.999...=1 is crucially dependent upon which number line is presupposed. — alan1000

No, it depends on what is meant by '...'. In ordinary mathematics, '...' in that context refers to the limit of a certain sequence, and we prove that that limit is 1.

in the (classical) real number line, 0.999... is the largest real number which is less than 1 — alan1000

Wrong. In classical mathematics, '.9...' is notation for the limit of a certain sequence, and that limit is proven to be 1.

Cantor's Diagonal Argument certainly seems to support this interpretation — alan1000

Cantor's argument has nothing to do with it. They are different matters.

Abraham Robinson — alan1000

Robinson came up with non-standard analysis. That is a different context.

.9999... = x
9.9999... = 10x
10x-x = 9.999... - .999...
9x = 9
x = 1 — flannel jesus

That's not a proof. It's handwaving by using an undefined operation of subtraction involving infinite sequences. Actual proofs are available though.

I think it's fair to say that's not a satisfactory proof because of infinite sequences, but this is absolutely not what hand waving looks like. Whether it's right or wrong, it is an explicit attempt to work through reasoning step by step - maybe that reasoning fails, but it's not handwaving.

It's handwaving. The argument invokes an utterly undefined notion. It's a garbage argument as far as mathematics goes. And it doesn't even have explanatory value, since it merely defers the question of what '...' means to the question of what subtraction on infinite sequences means.

Here's a proof:

Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j).

Let f(k) = SUM(j = 1 to k) 9/(10^j).

Show that lim(k = 1 to inf) f(k) = 1.

That is, show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e.

First, by induction on k, we show that, for all k, 1 - f(k) = 1/(10^k).

Base step: If k = 1, then 1 - f(k) = 1/10 = 1(10^k).

Inductive hypothesis: 1 - f(k) = 1/(10^k).

Show that 1 - f(k+1) = 1/(10^(k+1)).

1 - f(k+1) = 1 - (f(k) + 9/(10^(k+1)) = 1 - f(k) - 9/(10^(k+1)).

By the inductive hypothesis, 1 - f(k) - 9/(10^(k+1)) = 1/(10^k) - 9/(10^(k+1)).

Since 1/(10^k) - 9/(10^(k+1)) = 1/(10^(k+1)), we have 1 - f(k+1) = 1/(10^(k+1)).

So by induction, for all k, 1 - f(k) = 1/(10^k).

Let e > 0. Then there exists n such that, 1/(10^n) < e.

For all k > n, 1/(10^k) < 1/(10^n).

So, |1 - f(k)| = 1 - f(k) = 1/(10^k) < 1/(10^n).

"The set of all sets that do not contain themselves". Obviously this top set could not self reference. I would say the same of Godel — Gregory

"Applying the rules consistently breaks down. Therefore we do not apply the rules consistently".

If you apply the rules of naive set theory - those rules lead to a contradiction. Therefore the rules of set theory cannot possibly be correct (The Principle of Explosion).

When you have to make an exception to the rules in order for things to work - your rules don't work.

As such - I don't think your comparison to an inconsistent set theory and its ad hoc fix is very helpful.

More pertinently - Godel was specifically working within the rules of axiomatic mathematics and exploring the limits of those rules. If he had to step outside those rules then the whole point of the exercise collapses.

The whole point of Godel's incompetenesses is: Given premise; what can we say?

For the incompleteness theorems to exclude themselves would be to disregard the whole point of the exercise in the first place.

It seems to me that geometry/space is what has presente a foundation for all mathematics. — Gregory

Yes-ish.

You have to exist in order to have a concept of mathematics. The universe has to exist for you to exist for you to conceive of mathematics.

So - yes - absolutely - mathematics is founded on the existence of the universe.

The trouble is that a full understanding of mathematics requires a full understanding of the universe as a pre-requisite.

Your concept of numbers derives from your experience of the world around you.

But no-one can define the universe in a fixed, objective manner.

Axioms

Given a set of axioms we can create an axiomatic system.

But...

In order to uniquely define a set of axioms we need a set of instructions that describes how axioms should be interpreted: axioms^2

In order to accurately interpret axioms^2 we need a set of instructions the describe how axioms^2 should be interpreted: axioms^3.

...

Etc, etc and also etc.

So - yes - we do in fact take the universe as a foundation and explore that foundation. But we can't say anything definitive about that foundation. Consequently we cannot say anything definitive about anything derived from that foundation.

So - we are free to propose the existence of the real number line - but we cannot say anything meaningful about it. Any definition faces the problem of axioms - infinite regression (or a closed loop of A defines B and B defines A).

No matter how hard we try - we are only ever able to describe the relationships of X, never X itself.

As with Zeno's paradoxes where we see space dissolve into nothing (or parmendian pure being), numbers must have a basic unity that holds them from infinite divisione. If we have 1, then we have 2 halfs, which each is one, so 1 is two. This can go on forever- as with divisione of a line. Numbers are synthetic (Kant)and nah platonic (Plato). A number is not a set. 7 is not the set of 7 ones. Sets are applied by us TO numbers which WE can choose how to group — Gregory

In mathematics - a paradox (inconsistency) demonstrates a faulty set of axioms.

Zeno's paradox demonstrates that some assumption (such as the continuous nature of space) is mistaken.

I would argue that zeno's paradox is a demonstration that space is not, in fact, continuous. That space cannot be infinitely divided - just as we currently believe matter cannot be infinitely divided (c.f. electron is a fundamental particle).

In mathematics - a paradox (inconsistency) demonstrates a faulty set of axioms — Treatid

Not necessarily. The Diagonal paradox can be extended to a sequence of smooth curves that converges to a limit curve in the complex plane in which the disparity of lengths is infinite. There is no argument I have heard of that implies fundamental axioms of the real (and complex) numbers is at fault. I seem to recall Aristotle was aware of this discrepancy of lengths.

Why can't we just choose to say the set of all sets that do not contains themselves is the highest in order and so is not included in itself? What in math or language requires that include itself in itself?

And i also am curious why Godel thought self reference a necessary step in mathematics instead of being contingent on our will

Maybe i am an intuitionalist