# Is 1 = 0.999... ? Really?

Also, you say there is more than one Infinity or different sizes of Infinity. How?

123456…m is the same as 999…km and 888…cm and 98873285438534789…mm

Can you show me how you can have an ???..m/km/mm/cm such that ???.. is greater than the above? You cannot. So why does mainstream maths believe in different sizes of Infinity? Given that you describe yourself as a pure reasoner, why do you??

It is because you leave out often critical words such as - “the”, “a”, “those”, “portions of”, “a part of”, “a measure of”, “includes”, and many similar clarifying details that prevent vaguery.

I am certainly not a grammar expert and I am certain that I am often grammatically incorrect but when a grammar error leads to something blatantly illogical, I certainly try to catch and correct. Such as -

That is a blatant violation of logic. What that statement says to the purely logical is -

• Existence is the exact identical thing as “immeasurably large” and also
• The immeasurably large is exactly the same thing a “immeasurably small”

So an easy conclusion is that the set of natural numbers is Existence itself. It isn’t that Existence INCLUDES the natural number set but rather that the natural number set IS Existence - implying that anything that isn’t the natural number set is not Existence. But then the set of all odd numbers (also infinite) is also Existence itself. So the conclusion must be that the set of all natural numbers is the exact same as the set of all odd numbers. You don’t see the “paradox” in that?

That is what happens when you leave out those clarifying words that address important details.

It is absolutely false that existence is exactly the same as the set of all natural numbers (which is infinite). Yet that is what you said with “Existence ( = ) Infinite”.

What you could have said is -

• Existence includes all that is infinite and also
• Existence includes all that is infinitesimal
• Or
• Existence is all that is infinite, infinitesimal, and everything between

Can’t you see the difference?

There are probably many things that could be described that way - beauty, love, wisdom, intelligence, virtue, - maths,… And poetically, throughout poetry, they often are described that way so as to add a sense of magnificence - a touch of evangelizing.

You are right except for one small difference involving that question about how you discern what does or does not exist. It is easy to discern what does or does not have affect.

Note how you stated that in blue - “E is infinite in measure”. You didn’t say that “E = Infinite”. Those are distinctly different statements - the first being true - the second being false.

Now that I can interpret most of what you are trying to say, I can agree with your intent - but not the way you word it. Your wording defiles logic and reasoning - not your intent. Logic IS about words and the concepts they uniquely represent (A = ANOT A = A+3 - regardless of whatever “A” represents).

I don’t have significant issue with any of that.

I think in Logic they call that “a non-sequitur” meaning that the premise didn’t actually lead to the conclusion even though the conclusion might be right. And also “Existence exists” is a bit meaningless - “tautological” - vacuous - “How do you know it is red? - Because it is red!” - “Why are you mad? - Because I am mad” - “Did you notice that black is black?”

So it should be equally absurd that existence involves size - it either exists or it doesn’t - it is either real or it isn’t - size has nothing to do with it. What happened to all that talk about infinite and infinitesimal?

No one said anything about being real or not. No matter the density, if it exists at all, it is obviously real.

Density merely adds another measure. An existing thing can have spacial size and also have density. And it cannot be void of either.

If you mean “infinite size” there is certainly a thing with infinite size. But that same thing is NOT infinitely dense. In fact according to science that thing is very low density - mostly “empty” space (although we know that even “empty” space isn’t totally empty).

So unicorns exist? Everything you can imagine exists? If I imagine a man standing behind you with a knife about to stab you - he exists?

That could lead to some extremely serious court cases.

To me, the Infinite = the Infinite. The set of all natural numbers = the set of all natural numbers. The Infinite contains/encompasses the set of all natural numbers. It is not ‘all natural numbers’. I think it is the only thing/set that contains all natural numbers. Being all things and encompassing all things are two different truths. Actual Infinity is that measure which Cantor seemed to believe to be non-existent, hence his assertion that there is no such thing as the greatest cardinal number. What then is the number Infinity? We cannot count to the number Infinity, but it is still a number we are trying to count towards. We are not counting to red are we?

Also, Cantor’s assertion that there is no set of all cardinalities (if Wikipedia is correct). The Infinite is one thing/set/number that encompasses ALL numbers (including Itself). Much like how the meaning ‘meaningful’ is just one meaning. Yet, all meanings are members of it. In other words, all meanings are meaningful despite only one meaning actually meaning ‘meaningful’. All meanings are meaningful, despite only one meaning being a member of itself. Despite only one meaning meaning meaning.

There’s a reason why I did not say the above. I wanted it to be clear that there is only one Infinitesimal. What you say above may mean this, but it can also mean that there is more than one Infinitesimal. I am not clear on your position regarding this. It is paradoxical for there to be more than one Infinitesimal. Agree or disagree? If disagree, what separates one Infinitesimal from another?

Also, when I talk about that which is Infinitesimal in measure, can I be talking about anything other than Existence?

If you see a unicorn in your dreams, what does that mean to you? Surely at the very least it means that your dream exists. The unicorn in your dream exists. Existence does not just encompass one form or degree of reality. Thoughts and dreams do not pop in and out of Existence. Nor are they nothing/non-existent. They all exist in Existence. What thoughts or hypothetical possibilities you focus on, access, or experience, or how you experience them in Existence, is another matter. Because Existence Is Infinite, the hypothetical possibilities are Infinite.

All meaningful things you can imagine exist because all hypothetical possibilities truly exist (of which the number is Infinite). As in hypothetical possibilities truly are hypothetical possibilities. To say otherwise is to be contradictory because it amounts to saying x is a hypothetical possibility but it is not an existing hypothetical possibility. Whether or not a hypothetical possibility will attain reality in relation the reality that we label ‘our real reality’ or the reality that we label ‘dream’, is another matter. The potential backstabbing man you imagine is a hypothetical possibility. If it is not an absurdity/hypothetical impossibility, then it can attain reality in relation to me. It is the Nature of Existence that an Infinite number of hypothetical possibilities exist. It is the nature of God that God Is Omnipotent.

CR,

You’re not stupid, but then again, you are.

I can imagine every being in existence being in the worst possible hell forever. Is that what’s occurring right now? No.

You’re making me tired trying to explain existence to you. I just want to tell you to shut the fuck up and listen. You’re like the drunk dude at the bar who gets 43rd because he’s such an asshole.

We access imaginary realms to live, that doesn’t mean it’s manifested. This is a good thing about existence.

Modal realism is a shitstorm.

I understand that you’re not intelligent, wise or a good person.

Existence as we currently experience is shit. You, are a sycophant …

By the way, if you being a sycophant works out well for you, it’s because of the extremely hard work of people like me, not because of you… you’re an infant in this cosmos.

I don’t even know what that means. I tried to guess earlier - but - who knows.

That depends on how you answer the question that I asked about how to discern what exists and what doesn’t. If anything I imagine exists, then what might be infinitesimal is up to my imagination.

I don’t think so. The unicorn in my dream is only an idea - no different than imagining the man standing behind you with a knife. The imagining process exists - the subject of the imagination doesn’t - else it wouldn’t be called “only imagined”.

I guess what you are talking about is that separation between existing as a divine object (concept, idea, or imagined) and existing as a physical object. James notes those as - “The Two Realms of Existence” - “Conceptual Existence” and “Physical Existence” and says that Plato had that same distinction - “Divine and Mortal” (I suppose that is true).

There clearly should be a distinction between a perfect circle and a physical circle because the physical universe cannot ever have a perfect circle - or any other geometric form. Ideas exist AS IDEAS (Biblically as “Angels”). Angels are not mortals (obviously you are not Catholic ).

That is a claim that you are making - not a reasoning. “Hypothetical” doesn’t mean the same as “certainly existing”. The hypothetical refers merely to something guessed or supposed but perhaps not at all real - “Hypothetically I could balance a bowling ball on my head” - that doesn’t mean that I really can - only that it might be possible - a guess about what is real - and in reality perhaps completely impossible. So I cannot agree that every hypothetical exists or is real.

If it must attain reality then it certainly doesn’t already have it. And it can only attain reality if other circumstances allow for it. So until it is actual - it isn’t there.

Like the dream - the guess itself exists as a guess - a thought. But the conclusion of the guess doesn’t exist just because it was guessed. If I guess that I have already won \$million is it really there? It MIGHT be - but it might NOT be. So we have -

• It exists
• It does not exist
• I don’t know if it exists

Where do you place the “I don’t know” category?

No. It isn’t.

1 is always one, and 0.999… is always 0.999…

Such an equation serves only mathematical requirements.

Alright, let’s focus on one point at a time. Let’s discuss the Infinitesimal, then the Infinite, and then after that, let’s discuss what it is to be real/exist and what it is to be as real as us (exist in the manner that we do), but in another thread. For now, just the Infinitesimal:

What the semantic of triangle is, is not up to your imagination. Similarly, what the Infinitesimal is, is not up to your imagination. What you imagine, is not up to your imagination. What you want/choose to imagine, is up to you.

Reflect on the Infinitesimal and tell me where you stand on the following:

It is paradoxical for there to be more than one Infinitesimal Thing. Agree or disagree or unknown or can’t make sense of the statement? If disagree, what separates one Infinitesimal from another? If unknown, then try to reflect on what it is for one infinitesimal to be here, and another infinitesimal to be 5cm away. What separates them?

If you can’t make sense of the statement, then focus purely on the semantic of ‘infinitesimal’. Does something infinitesimal exist? Is it absurd or possible for more than one of it to exist?

Obsrvr,

I’ve been in a bad mood as of late, thanks for making it civil.

I honestly get tired of this crap. Patience of a saint right? Not much for me in my old age anymore.

I’m only 44. It feels like trillions of years.

Can’t make sense of the statement.

An infinitesimal of what?

Any thing.

You name it.

Really? Ok. ‘An infinitesimal of what?’, is an absurd question. This is because Existence is the only existing thing that Is Infinite and Infinitesimal in measure (depending on from which angle you want to try to look at It).

I am unclear with regards to whether you believe in an existing infinitesimal thing or not. If I remember right, you said something like affectance fills every infinitesimal of space. So do you think there is more than one existing infinitesimal? If yes, what separates one existing infinitesimal from another?

Given the above, it might also it might be worth drawing your attention to the following:

999…m and 23234892…mm and 111…km all denote the same Measure. The Infinite. You say Infinity comes in various sizes. Can you show this to me? And can you show me how you can have more than one Infinity such that one Infinity is clearly distinct from another infinity?

1.999… and 1.1231122375748… and 8.999… all denote the same Measure/Thing. The Infinitesimal. You have not said the Infinitesimal comes in various sizes. But your words suggest that there can be more than one infinitesimal. Can you see how 1.999… and 8.999… denote the same Thing/Measure? As in can you see how 1.999… is not greater than 8.999…?

Yes, 1.999 is smaller than 8.999. But what about 1.999… and 8.999…? Is it not the case that 1.999… and 8.999… is just another way of saying/highlighting the Measure Infinitesimal? What separates/distinguishes 1.999… from 8.999…? Nothing. What separates 8.999 from 8.9999 from 8.999…9 from 9? The Infinitesimal. What separates 8 from 9? The Infinitesimal. What is between 8 and 9? An endless number of numbers and the Infinitesimal. The Infinitesimal Separates all these numbers from one another. What separates 9 from 9? Nothing. 9 IS 9. Just as Infinitesimal Is Infinitesimal.

No. It is not.

If you wanted to highlight that measure, you would say -
2.0 - 1.999… = 1 infinitesimal.

And -

The infinitesimal just above 2.0 is a different infinitesimal than the infinitesimal just above 1.0. And they are separated by an infinity of other infinitesimals.

I think I am increasing in clarity. I thank God for this, and I thank you for taking part in this discussion.

I am clear on the fact that 1.999 is 1.999 in measure. And that 2 - 1.999 = 0.001. But I now think that I am also clear on the following:

There is no difference between 0.111…m and 1.999…km and 999…mm.

If you see a difference in semantics (not letters and symbols), can you meaningfully and non-paradoxically convey it to me? With that being said:

Is 1.999… not an infinitesimal measure? If it is not an infinitesimal measure, is it not an infinite measure? If so, how can it possibly be that 1 minus infinitesimal/infinite = 1 infinitesimal/infinite?

How do you take away an infinitesimal or an infiniteth from something, and still be left with the exact same measure? You either cannot take away an infinitesimal, or you take away an infinitesimal and are not left with the same measure. The latter is absurd (much like negating Existence is absurd), as is any alternative to the former. With pure reason in mind, such that it is not sacrificed in the name of mathematics, or what famous people say, or what our predecessors said, or religion, is what I am saying correct or incorrect?

Zookers. I don’t think I can.

“0.111… meters is the exact same as 1.999… kilometers which is the exact same measure as 9999… millimeters.”

What?!

How can you not see that -* 0.111… is close to being merely 0.1

• 1.999… is close to being merely 2.0 and
• 999… is infinitely large?

Are you perhaps thinking that the symbol “…” relates to something infinite and not seeing that it is actually only referring to the infinity of digits and NOT to the value those digits represent? Every digit to the right of the decimal is a diminishing measure, not increasing measure - each represents something smaller - and smaller - and smaller - almost to zero.

If not that, I have no idea why you would think those numbers are anywhere near being “no different”.

Certainly not. It is a measure that involves and infinity of digits to represent a measure only 1 infinitesimal from being 2.0. It is the same as 2.0 - 1 infinitesimal - a measure close to being 2.0 - certainly not infinitesimal (unless the number of your eyes is infinitesimal ).

Not unless and until 2.0 is deemed infinite.

Ok, so before, I believed that 999…m and 888…km were the exact same measure (I still believe this). I also believed that 1.999…m and 999…cm were different measures purely because of the decimal point (I no longer believe this). I think this is how you are seeing it now but I will verify before going further:

1. Do you agree that 999…m and 888…km are the exact same measure? If no, which is bigger/larger?

2. Do you agree that 0.999…m and 0.888…m are the exact same measure? If no, which is larger/bigger or shorter/smaller? Is your answer consistent with 1?

Which is closer to infinity? 111… or 999…? Which is closer to 2? 1.111… or 1.999… or 1.159… ? Isn’t what you say in the quote above like saying: a number only 1 number away from being Infinity? 1.999… is no closer to 2 than 1.87652… or 1.111… are. 1.9 is closer to 2 than 1.8.

2 is no closer to 1.999… than 1 is closer to 1.111… AND 1 is no closer to 1.111… than 2 is closer to 1.111… They are both equally close (if they are at all close) Why would 1.999… be any closer to 2 than 1.111… be?

Almost to zero or going towards almost to infinity/infinitesimal? But it’s not just going towards infinitesimal. It is an infinitesimal measure. If you view 999…m or 888…km as being infinitely large (an infinite measure…or simply infinity), then you must view 0.999…m and 0.888…m as either infinitely small/short or infinitely large/long (an infinite measure…or simply infinity). You cannot take infinity out of the equation when … occurs. So where do you stand? Infinitely small or infinitely large? Or nothing to do with infinity?

Bearing in mind there is a difference in meaning between ‘to infinite/infinitesimal’ and ‘is infinite/infintesimal’. Or a difference between ‘to infinity’ and ‘is infinity’, can you answer the following:

1. Do you agree that both 1.999…km and 2.999…mm are either infinitely small or infinitely large? If they are not infinitely anything, what are they? And if they are infinitely something, which is larger?

If they are both infinitely large/small, which is larger/smaller?

Do you see my point? How can that which is infinitely large, be larger/smaller than that which is infinitely large? And how do you know if 999…mm or 0.009…km or 99.999…m is infinitely large or infinitely small? Shouldn’t you just say they all denote the Infinite or the Infinitesimal or Infinity?

Hi there @Certainly real. By way of introduction, I’m a veteran of this thread. I participated heavily several years ago, and a little bit last year. I noticed some basic misunderstandings about numbers that I thought I might be able to clarify. My background is in math, and I have enough interest in philosophy to relate to some of the non-mathematical concerns about the mysterious .999… = 1. Let me just say one thing up front. That equation is a valid theorem in standard mathematics. Whether you accept that as the answer; or whether you reject it on philosophical reasons, either way, it is a fact that IF you accept the basic rules of math, THEN .999… = 1 follows as a theorem. For example it’s proved to be true in freshman calculus, as an example of a geometric series.

But I don’t want to talk about that today. I want to simply clarify some basic aspects of numbers and their representations.

1. All counting numbers: 1, 2, 3, 4, …, have finite length representations. The rule to generate the counting numbers (also known as: the natural numbers, the positive integers, the positive whole numbers) is that you start from 1 (or sometimes 0); and to get the next number you add 1.

Given that rule, you can see that if I have some number like 88888888, the next one is 88888889. And if I have, say, 99999999, the next one is 100000000. It rolls over like the odometer in your car. The point is that every single number you get by the “add 1” rule has a finite length.

Therefore expressions such as 888888888… or 99999…, where the ‘…’ means that the pattern continues forever, have no meaning in math. So it makes no sense to ask about them. You can’t ask which is bigger, because neither of them has any mathematical meaning. All whole numbers, all counting numbers, have a finite-length representation.

1. When you talk about m or km, that adds confusion, because we are not talking about anything in the real world. The question, is .999… = 1 in the real world is meaningless. The fact is that we can’t measure anything that precisely. For one thing, our measuring instruments are imprecise.

For example our most accurate electron microscopes have a resolution of about 1 picometer, that’s (10^{-12}) meters. That’s .000000000001 meters. And even worse, all our theories of physics break down below the Planck length. That is, there is a physical length below which it doesn’t even make sense to talk about anything being smaller. Not that there is or isn’t anything down there; it’s just that our physics doesn’t allow us to sensibly discuss the matter.

So when you talk about what is .999… meters, I have to say that the notion is meaningless. We can not measure or even sensibly discuss anything smaller than about 30 or so 9’s. Smaller than that, and physics itself doesn’t work.

I hope we can agree that questions of physical measurement involving m or km are irrelevant here. All we are concerned about is the purely mathematical meaning of numeric expressions.

To repeat this, because it’s important: We are discussing pure abstract math and NOT physics. Even though .999… = 1 is true in math, I would NOT say it’s true in the physical world. It’s meaningless in the physical world, because we can’t measure anything that finely. Perhaps this is a frequent point of confusion. We’re not talking about the real world here, only pure abstract math.

1. Now to the crux of the matter. I think you might be a little unclear about how decimals work. You have stated that an expression such as .1111… is infinite, or infinitesimal. It’s not. In fact .111… is precisely equal to the familiar fraction (\frac{1}{9}).

Before getting into that, though, let’s just review how decimals work. If you have a decimal like 0.1, the digit position to the right of the decimal point stands for the number of tenths. So 0.1 means, literally, (\frac{1}{10}).

The second digit to the right tells you how many hundredths there are. So 0.11 means 1 tenth plus 1 one hundredth, or (\frac{1}{10} + \frac{1}{100}), which is equal to (\frac{11}{100}), or a little bit less than (\frac{1}{9}).

If we add another digit, that’s the number of thousandths. So 0.111 means (\frac{1}{10} + \frac{1}{100} + \frac{1}{1000}), which adds up to (\frac{111}{1000}), which is a little bit smaller than 1/9.

With this understanding, I hope you can see that a number like 0.111111111111111 is even closer to, and still just a little bit less than, 1/9. We don’t even have to believe in infinitely long decimal expressions to see that for any finite number of 1’s, the number 0.1111…1111 is a finite number, larger than 0 and a tiny bit smaller than 1/9, as long as there are only finitely many decimal positions containing ‘1’. If you then allow expressions with infinitely many decimal places, we can show that 0.111… is exactly equal to the familiar fraction 1/9.

Even though there are infinitely many digits, the number that’s represented is just a familiar fraction between 0 and 1, namely 1/9. Like cutting up a pizza into nine slices. Each slice is 1/9. Of course we could never physically measure 0.111111…, but that is no concern of ours. We are in the realm of pure abstract mathematics, and NOT the physical world of solid things around us.

I hope this is clear so far. I want to just leave it at this and not talk about infinitesimals yet, because the concept of decimal notation is important.

Here’s another example. Say we cut our pizza into the more usual 8 pieces. What is that in decimal notation? Well, 1/8 = 125/1000, which turns out to be 0.125. That’s 1 tenth, 2 hundredths, and 5 thousandths. That’s how decimal notation works.

To directly respond to questions you’ve posed: The number 1.11111… is exactly 1 and 1/9, or 10/9. And the number 2.999… is exactly equal to 3. But even if you don’t believe that, you have to at least agree that it’s larger than 2 and certainly no larger than 3. It’s not infinitesimal and it’s not infinite.

Let me know if this is helpful so far.

This is really strange. I have never seen anything like this. You have introduced a variety of issues so let’s see if we can deal with one at a time.

The four numbers that you have pointed out are only common in that they each USE an infinity of DIGITS to describe themselves. Only two of them are related to an infinite VALUE (those in (1)). Don’t confuse the use of digits with values that the digits represent. An infinity of digits can be used to describe a very small value.

That is the first issue so do you understand and agree with that much?

I can understand how 0.999… = 1 Infinity/Infinitesimal/Infinite/Existence. But I cannot understand how 0.999… can = 1 pie or 1 cake.

We are in agreement on this. If x forever expands, then x is forever finite but always expanding in its finiteness. It always has an end that it is surpassing. It also had beginning, therefore it will always have a beginning. This is not the same as that which truly has no end and no beginning. That which is expanding without end (to Infinity) is not the same as that which IS Infinite (that which cannot expand, that which IS Infinite)

Ok, for the sake of argument, when discussing Infinity, forget numbers. Do the following words have any semantical value: ‘Infinitesimal’ ‘Infinite’

If they are meaningful, are we in agreement that there is no number greater than Infinity. If you say Infinity is not a number, then how do you respond to the following:

We try to count to x. We try to count to 10. We try to count for 10 minutes. We try to count forever. We try to count to Infinity. Is infinity in the previous sentence not a number that we are trying to count to? If it is not a number, what is it that we are trying to count to? Note that if I try to count to hsjg, that’s meaningless. This is not the same as me trying to count to Infinity. You cannot count at all when you try to count to hsjg because if x tells you to count to hsjg, you wouldn’t know what you are counting to. If x told you to just keep counting, you would start counting, but again you wouldn’t know what you are counting to. If x told you to count endlessly, then you know he wants you to count to infinity. But you can’t. This does not mean that you did not understand what he meant. Therefore, clearly, you understand the meaning of Infinity. You can count when you try to count to Infinity, but you know you will never reach that number.

My focus is not on what we are empirically capable of. My focus is on what we are a priori aware of. My focus is on the semantics that we are aware of. Infinity, Infinitesimal, and Infinite have a clear semantical value for me. They are not meaningless to me like qwer or 10th dimension.

So are you saying that Infinity/Infinitesimal/Infinite are all as meaningless to you as Hfdshjsf and 100th sense or 10th dimension? They are not as meaningless to me. I can have a coherent understanding of Existence without the ‘semantic’ of 199th sense. But I cannot have a coherent understanding of Existence without the semantic of Infinity because without Infinity, I would have to accept that Existence came from non-existence (which is absurd). In science, we do not accept absurdities/paradoxes. We reformulate our theories and shift paradigms by formulating completely new theories if necessary, to avoid inconsistencies/paradoxes/absurdities in our theory. Existence coming from non-existence is an absurdity. Therefore, we must Existence has always existed and will always exist. Hence the clear meaningfulness and necessity of Infinity in any given understanding of our world and Existence.

Pure reason, of which philosophy and math are more committed to than physics, dictate the limits of physics. If physics ever asserts that Existence came from non-existence, it would be contradicting pure reason. It would be rejected on the ground of being absurd/paradoxical. If you tell me that the Infinite is not at least as real as you and me, you’d be paradoxical for the following reason:

Existence is at least as real as you and me. Existence Is necessarily Actually Infinite. Therefore that which is Actually Infinite is necessarily at least as real as you and me.

I think I completely understand why you would say this. I appreciate your efforts to educate me on this. But this part with the fractions and the decimals, I’m ok with. I can appreciate that what I am proposing may seem that I am not ok with 0.1 equaling 1/10, but I am. My belief is that 111… is the exact same measure as 168… and 0.111… 0.000121…

Where … occurs, as in where we imply that the numbers truly go on forever, then the measure that we are highlighting is Infinity is it not? 11124…56 is a finite measure. 1124… is not a finite measure is it?

Similarly 0.111…234 is a finite measure. 0.111234… is not a finite measure is it?

Again, forget notions of what we are able to empirically observe. I am not concerned with empirical observations. I am concerned with pure reason. And bear in mind that we will never be able to empirically verify Existence Being Infinite. But we are rationally obliged to accept that Existence Is Infinite. Again, when there are paradoxes in any given theory, there are paradigm shifts, reformulation of the theory, or a completely new theory. So, with that in mind, can you see how Existence (the Actually Infinite), is necessarily at least as real as us?

1.11111… is what separates 1 from 1/9 and 1 from 1 and 1/9th. 1 does not equal 1 and 1/9 precisely because 1 and 1/9 is a finite measure. 1.1111… is not a finite measure.

Ok sure.

Yes. No disagreements here.

Despite agreeing that all 4 number use an infinity of digits, we disagree here.

Ok. You say a very small value. Can there be more than one very small value when we are talking about a number that consists of an infinity of digits?

Also, can there be more than one very large value when we are talking about a number that consists of an infinity of digits?

Would you describe 999…km as infinitely large/small? Would you describe 99.999…km as infinitely large/small? Would you describe 0.999…mm as infinitely large/small?