Is 1 = 0.999... ? Really?

[Magnus Anderson]

Every partial product is greater than zero, yes - because it's only partial! Thus the same logic carried over to any "non-partial-product" of some kind of "infinite product" is invalid. Again - poor logic skills on your part.

Every partial product of \(0 \times 0 \times 0 \times \cdots\) is equal to \(0\) and that's precisely why its result is equal to \(0\).

The same does not apply to \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots\).

The fact that it's impossible to narrow down any possible value should ring alarm bells that this "proof that the gap exists" is flawed.

Who told you it's impossible? I said it's unnecessary.

The reason we care about the poor guy

It's because you're a confused and at the same time conceited mind (hence your defensiveness.)

I mean, it literally necessarily is the case by definition - you even accepted the fact that they are identical sets insofar as they have one-to-one correspondence.

I accepted that \(A\) and \(B\) are identical, not that \(B\) and \(B'\) are identical.

Take a break. Relax. Mull it all over - it'll be good for you.

Yes, it will be good for you. If anything, you'll be less focused on who you are in relation to other people and more focused on the actual subject.

Can you do this for us, please?

Or do you REALLY have to tell us how great you are each time someone says something negative about you?

[Magnus Anderson]

It has to be a quantity smaller than itself to be valid, which is invalid, making it invalid either way. It's a contradiction in notation either way - it's logically not a possible quantity.

\(0.\dot01\) is a contradiction, but even if what it's trying to represent wasn't a "fully bounded boundlessness", it's a quantity that is always smaller than it is: it always has to be divided more to suffice as small enough. Logically the only quantity that is small enough to not be divisible further is zero, which not incidentally is the limit of what this gap has to be. So even though we can't get to the limit, we can confirm through logic that the answer is in fact zero, as though we actually could get there by means of the limit.

\(0.\dot01\) is a number that is greater than \(0\) but smaller than every number of the form \(\frac{1}{10^n}\) where \(n \in N\).

That's what \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots\) means. It is telling you "This is a number greater than \(0\) but less than \(0.1\) and \(0.01\) and \(0.001\) and \(0.0001\) and so on".

You can say "But there is no real number that can represent such a quantity!" to which I can respond with "But the set of real numbers is not the set of all quantities." I can then add that numbers are merely symbols invented by humans in order to represent quantities that are of relevance to them. They are not quantities themselves. Symbols are few and far between, quantities are numerous. And when it comes to quantities, they can be as small or as big as you want them to be. There is no limit.

All in all, all you have to do is understand what that infinite product represents. But you're refusing to do so, obviously because you have your own agenda -- it's in your interest to misinterpret it.

What's certain is that \(0.\dot01\) is not a number that is less than itself. That's your own invention.

[Magnus Anderson]

A contradiction would be a statement such as "A number that is greater than \(1\) but less than \(\frac{1}{2}\)". That would be a contradiction because we said that a number is greater than \(1\) (P) and not greater than \(1\) (not P.)

No such contradiction is committed when we say that \(0.\dot01\) is a number that is greater than \(0\) but smaller than every number of the form \(\frac{1}{10^n}\) where \(n \in N\). That's because every number that we say is greater than \(0.\dot01\) is also greater than \(0\).

Another contradiction would be a statement such as "A number that is greater than \(1\) but less than \(1\)". A number can't be both greater than and less than one and the same value. But that's not what we're doing either.

Of course, it is possible that we are contradicting ourselves in other ways, by contradicting other statements that we've previously made, but the fact is, there are no other statements we've made.

The problem with Silhouette is that he's starting with the premise that there are no numbers smaller than all real numbers. If you START with such a premise, and then you say there is such a number, that would be a logical contradiction. But noone other than him (and those who think like him) start with such a premise.

Certainly not me.

And since this is MATHEMATICS and not EMPIRICAL SCIENCE the only thing we are interested in is LOGICAL CONSISTENCY. You can't say that some premises are true and others are not. The only thing that matters is consistency: making sure you are not contradicting what you previously stated.

Why would anyone start with a premise that there are no numbers smaller than every real number?

There is LITERALLY no limit to how small a number can be (in your head, of course.) No matter how small a number is, you can always imagine a smaller number. There is no limit whatsoever. All it takes is a statement such as "Here's a new number N that is smaller than every number that is currently familiar to me!" You don't have to imagine it or visualize it. All it takes is to state its relation to other numbers.

So what's the point of limiting yourself to real numbers?

Inability to think abstractly?

[Silhouette]

You're resorting to nonsense about conceptual matters being separate from empirical matters because deep down you recognise the only "traction" you have left on this issue is to insist that your argument doesn't "need" to be testable.

What makes you think that only empirical claims are testable?

Nothing, because I explicitly said that "Most of my objections are actually logical ones, and occasionally I back them up with empirical objections - either way, in order for you to forward your conceptual propositions as knowledge, they need to be testable and tested."

I never said only empirical claims are testable - you are making things up AGAIN.

I never said nor implied that this thread is about untestable claims. I merely said that it's not about the kind of questions that you (and Gib) have been obsessing over ever since you joined this discussion.

This thread has nothing to do with science, epistemology and information theory. It has to do with mathematics and logic.

Unfortunately you don't even realise that you have been implying that this thread is about (your) untestable claims at multiple points. Every time you claim you don't "need" to prove your (il)logical claims, or that the logical arguments of others either don't exist when they clearly do, or you simply assert they're false when you notice them whilst at the same time accusing others of that very same fallacy that you're committing, as well as every time empirical tests are offered to help you understand your errors that you simply claim are irrelevant - you're arbitrarily invalidating the possibility for both logical and the empirical disproofs to apply to your attempted "proofs".

Basically you're constructing epistemological double standards for this thread to make it impossible for validity to prove you wrong whilst simultaneously dictating that invalidity can prove you right. In itself this is fallacious - "moving the goalposts". This is why you have no intellectual honesty, integrity or authenticity.

Homework for you: devise an argument that would successfully cause you to change your mind that isn't simply a reiteration of your own points and how you think they're right.
That is intellectual honesty - the ability to evaluate both sides of the story: the existence of which I've been trying to communicate to you all this time.
Obviously you've not been in higher education, or you'd be versed in this kind of thinking because that's what qualifies you for it and qualifies you to succeed in it. Either that or you completely disregard this kind of thinking especially for the internet - but from what I've seen I'm going to have to guess you never even made it, and perhaps realised it was never worth you trying. Must be the fault of the institutions though, right? Never yours.

Why does B' now have an extra 0?!

Because that's what you're doing. You're creating a new set, \(B'\), by adding \(0\) to it (the extra \(0\) you speak of) and then by adding all of the elements of the set \(A\) while relabelling them in order to make the resulting set \(B'\) look identical to \(B\).

\(B = \{0, 1, 2, 3, \dotso\}\)
\(B' = A \cup \{0_{new}\} = \{0_{new}, 0, 1, 2, 3, \dotso\}\)
\(B' = \{0, 1, 2, 3, 4, \dotso\}\)

In the final step, you're renaming \(0_{new}\) to \(0\), \(0\) to \(1\), \(1\) to \(2\), \(2\) to \(3\) and so on. That's how you create the illusion of one-to-one correspondence between the two sets.

You're correct that I intentionally add an extra \(0\) to \(B\).
You're incorrect that I'm "relabelling" the elements of set \(A\). I'm adding \(1\) to each of their elements and appending the result to the \(0\) that I first append to \(B'\). The fact that this makes them identical to \(B\) is no accident - this was intentional to show you that alternative set construction to end up with the same infinite set doesn't objectively change the set even if it now "looks like" injection when in fact set identity ensures bijection.

Obviously my efforts were yet again wasted.

Either way this simplifies as \(B' = \{0, 1, 2, 3, \dotso\}\), which is identical to both B and A as you accepted from the very start...

It does not simplify. When you write it like \(B' = \{0_{new}, 0, 1, 2, 3, \dotso\}\), it's pretty obvious that \(B'\) is greater than \(B\) because it has all of the elements that \(B\) has plus the new zero you added. But when you "simplify" it, you make it look as if they are identical. That's the trick.

So where's the extra element in \(B'\)? Well, \(0_{new}\) is the extra element. Don't conceal it and you'll see it.

I use the term "simplify" in the technical sense of making absolutely no change to the essence of the contents, not colloquially such that it potentially implies concealment or trickery. My terminology is wasted on you.

\(B'\) has all the elements that \(A\) has, with 1 added to all of them before they populate \(B'\) - to the preliminary ends of \(\{1, 2, 3, \dotso\}\)
That's why I add \(0\) first so that this set starting from 1 is appended to a set that has \(0\) already prepopulated within it.
I defined \(B'\) as \(\{0, 1, 2, 3, \dotso\}\) - stop pretending I didn't and actually read my explanation. No trick.

The only "trick" is if you mistake bijection for injection because of the intentionally alternative appearance of \(B'\)'s construction compared to \(B\) (and \(A\)) to conclude that \(B'\) now has more elements when if you follow each corresponding element to infinity it's literally impossible to find any injection.

I think you're done, don't you?

That would be you.

I wish it could be, but unfortunately I'm waiting on you. I won't give up until you've learned something: you the self-confessed non-mathematician professing mathematical expertise.

What would you most like to take from this thread? Victory? Self-improvement? Reconciliation? Teaching your expertise? Name it.

Every partial product is greater than zero, yes - because it's only partial! Thus the same logic carried over to any "non-partial-product" of some kind of "infinite product" is invalid. Again - poor logic skills on your part.

Every partial product of \(0 \times 0 \times 0 \times \cdots\) is equal to \(0\) and that's precisely why its result is equal to \(0\).

The same does not apply to \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots\).

I can repeat myself too if you want.

That's not getting us anywhere though is it.

I know every partial product is greater than 0. This just panders to your appeal to finitude that you keep denying you're doing.
INFINITES. We're dealing with them, not "partial, finitely somewhere along the way of an infinite series" which you're pretending still counts as valid for evaluating an infinite series.

The fact that it's impossible to narrow down any possible value should ring alarm bells that this "proof that the gap exists" is flawed.

Who told you it's impossible? I said it's unnecessary.

I showed you that it's impossible to narrow down any possible value above 0 to fill in any "gap" between \(1\) and \(0.\dot9\)
The only value that cannot be divided smaller, thus satisfying this "gap" is 0, otherwise it can always be made smaller.
I'm showing you that it's literally impossible. You're just saying it's unnecessary for it to be possible, but you're calling it "unnecessary to prove an exact amount".
The problem is that you don't appreciate the implications of your vagueness.
This is mathematics, not speculation.

The reason we care about the poor guy

It's because you're a confused and at the same time conceited mind (hence your defensiveness.)

You're picking up that I'm defending logic and reason huh? Sharp!
Defending myself though? Fuck that - I don't give a shit what anyone thinks of me unless it suffices as a means to the end of enabling the grander scheme of rationality to be upheld. I don't want that goal to elevate me or bring me down, it has nothing to do with me - rationality is objective and has nothing to do with the person who forwards it. It works regardless, independently. The only personal satisfaction that I take from philosophical discussion is in the aesthetics of truth. It's nice. Obviously you don't think so, but that's a result of your personal issues and has nothing to do with me personally. Call this kind of dissociation from conceit "conceit" all you like. I give zero shits.

I mean, it literally necessarily is the case by definition - you even accepted the fact that they are identical sets insofar as they have one-to-one correspondence.

I accepted that \(A\) and \(B\) are identical, not that \(B\) and \(B'\) are identical.

\(B'\) is defined to biject perfectly with \(B\), just with alternative construction to make it "look like" it's different to people fooled by the superficial.
As I keep saying and proving, that's you.

Take a break. Relax. Mull it all over - it'll be good for you.

Yes, it will be good for you. If anything, you'll be less focused on who you are in relation to other people and more focused on the actual subject.

Can you do this for us, please?

Or do you REALLY have to tell us how great you are each time someone says something negative about you?

I'm not giving up on you, Max.
No matter how frustrated it makes me.
I will make you learn even though I've been taking the entirely wrong approach of being honest with you about yourself this entire time. I should have been softer and less confrontational - I already knew that this just makes people act irrationally in exactly the same way that you are. I did the same thing with Urwrongx1000 but after a couple of years even he realised I was onto something.

I'm willing to wait that long for you, Magnus.

\(0.\dot01\) is a number that is greater than \(0\) but smaller than every number of the form \(\frac{1}{10^n}\) where \(n \in N\).

Oof! If you meant N as \(\Bbb{N}\) as the set of natural numbers you're in danger of starting to sound like you've been studying up on the subject!

Now we just need to eliminate your reliance on the invalid term \(0.\dot0{1}\) and maybe we can get somewhere?!

Perhaps if we formulate it like "Given the quantity \(x\) such that \(x=1-0.\dot9\), \(0<x<\frac1{10^n},\forall{n}\) where \(n\in\Bbb{N}\)" we might start sounding remotely close to professional.
But I'm just a mathematician, I'm not a professional one, so if anyone wants to correct me with any legitimacy - please feel free. I'm pretty sure there's a more succinct and correct, purely symbolic way of legitimately expressing that.

But this is just notation - the argument remains that whilst \(\forall{n}\) where \(n\in\Bbb{N}\), \(\frac1{10^n}-0\gt0\), \(\prod_{x=1}^\infty\frac1{10_x}\not\gt0\)

Basically "partial" products don't cut it by definition - like we already covered.

That's what \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots\) means. It is telling you "This is a number greater than \(0\) but less than \(0.1\) and \(0.01\) and \(0.001\) and \(0.0001\) and so on".

You can say "But there is no real number that can represent such a quantity!" to which I can respond with "But the set of real numbers is not the set of all quantities." I can then add that numbers are merely symbols invented by humans in order to represent quantities that are of relevance to them. They are not quantities themselves. Symbols are few and far between, quantities are numerous. And when it comes to quantities, they can be as small or as big as you want them to be. There is no limit.

All in all, all you have to do is understand what that infinite product represents. But you're refusing to do so, obviously because you have your own agenda -- it's in your interest to misinterpret it.

What's certain is that \(0.\dot01\) is not a number that is less than itself. That's your own invention.

I know what the expanded product of your infinite product is intended to represent.
I know that "there is no real number that can represent such a quantity" - as you say.
It is therefore your duty to comprehensively a valid population of quantities that satisfy the ability to exist between \(0\) and the difference between \(1-0.\dot9\)
You need to stop cowering from the formalisation of such a proof, apologising that it's "not necessary" by virtue that it "ought" to follow from what you regard to be a valid argument, which I have proven to be logically invalid.
You need to do better.
The Hyperreals already attempted this way before you, but as I've argued their conflation of the infinite with the finite is only a useful tool rather than truth.

It's not enough to say that "there should be some quantity there", you need to validly justify it - my "agenda" is with intellectual rigor ONLY. I don't give a rat's ass about anyone's excuses to get around that - I will hunt down your mistake and destroy it. It's fun and I enjoy the aesthetics of the result, but I don't care one jot what form it ultimately takes - even if it turns out you're ultimately right somehow.

It's a fact that \(0.\dot0{1}\) is a contradiction regardless of the fact that it was me who identified it. You cannot logically declare an infinite series of zeroes entirely bounded by a decimal point and a \(1\) because entirely bounded contradicts the boundless quantity of zeroes in between. Simple. Logic.

[gib]

\(10 \times \text{finite number} = \text{finite number}\)

But note that this isn't arithmetic. And neither is \(10 \times \infty = \infty\).

Exactly!

A couple of questions that need to be addressed in order to resolve the disagreement.

- What makes you think that \(0.9 + 0.09 + 0.009 + \cdots\) is a quantity and \(1 + 1 + 1 + \cdots\) is not?

- What is a quantity?

- Are you merely saying that \(0.9 + 0.09 + 0.009 + \cdots\) evaluates to a finite number whereas \(1 + 1 + 1 + \cdots\) does not? If so, what makes you think that \(0.9 + 0.09 + 0.009 + \cdots\) evaluates to a finite number? Most importantly, what does it mean for an infinite sum to evaluate to a finite number?

But I think I can predict your answers. Basically, what you're saying is:

\(0.9 + 0.09 + 0.009 + \cdots\) evaluates to a finite number and this simply means that its limit is a finite number.

\(1 + 1 + 1 + ...\) does not evaluate to a finite number because it is a divergent series (i.e. it has no limit.)

This is one of our main points of disagremeent: I think that the result of an infinite sum is not the same thing as its limit. The limit of an infinite sum is the value that it approaches but that it does not necessarily attain. The result of an infinite sum, on the other hand, is the value that it attains (i.e. it does not merely approach it, it actually attains it.)

\(5 + 0 + 0 + 0 + \cdots\) is an infinite sum that evaluates to \(5\). It does so because it attains \(5\). It does not merely approach it. Its limit is also \(5\).

\(0.9 + 0.09 + 0.009 + \cdots\) is an infinite sum that does not evaluate to a finite number. This is because there is no finite number it attains. Its limit, however, is \(1\) because it approaches \(1\).

In other words, neither \(0.9 + 0.09 + 0.009 + \cdots\) nor \(1 + 1 + 1 + \cdots\) evaluate to a finite number.

Ok, I see. That would make sense considering you don't agree that \(0.\dot9\) = 1. But I didn't know you thought it doesn't even equal a quantity. I thought, for sure whatever Magnus thinks \(0.\dot9\) does equal, he thinks it's a number. But I take it you think that because the infinite sum of 0.9 + 0.09 + 0.009 + ... never attains 1 (or just never completes), it doesn't even attain the status of being a number. Is that right?

I always thought that so long as it falls somewhere on the number line, it's a number:

... <-- -3 -- -2 -- -1 -- 0 -- 1 -- 2 -- 3 --> ...
...............................^ <-- \(0.\dot9\) falls here (either 1 or just before 1).

Infinity obviously falls outside the range of the number line (what I mean by "beyond numbers") and so it does not fall on the number line, and therefore isn't a number.

It's true that when a series approaches a limit, that limit can sometimes be attained and sometimes not, and the question of what kind of limit does \(0.\dot9\) approach is up for debate, but I always thought that the proof given at the wikipedia article settled the matter. If not, what would?

But what really interests me is the following: why does an infinite sum must evaluate to a finite number in order for us to be able to do arithmetic with it?

Because arithmetic deals with numbers and only numbers (which are, by definition, finite). It is the practice of calculating numbers. Throw anything else at it--like red, or cows, or shoes--and it makes no sense.

I don't think you can answer this question. You don't really know. And you don't really know because you did not reach this conclusion logically. You reached it by trying to justify a popular opinion. That's what I think.

You're calling the assertion that arithmetic only deals with finite numbers "popular opinion", huh? I'd call it self-evident, pretty close to axiomatic--that's just what arithmetic is.

You can ask me to justify my claims ad nausium, but we're getting pretty close to elementary principles here, which is to say rock bottom, end of the line. Soon it begins to be like asking: why does 1 + 1 = 2? Little children do this. The incessant chain of never-ending whys. Louis C K has a good bit on this:



The fact is, no one has an infinite chain of justifications stored up in their minds for all the things they believe (you're no exception). Press someone to justify their claims long enough, and you'll eventually get to the end of the line. That doesn't mean they don't know what they're talking about, it just means you've reach the fundamental level on which what they believe is based. At this point, it's just intuitive, self-evident, axiomatic. It's like trying to explain red to a man color blind from birth. If you find you can't do it, that doesn't mean you don't know what red is, it just means your knowledge of it is fundamental, basic, doesn't break down any further. At this point, you just know it or you don't.

I'm asking what it means to add something to an infinite set such that it's property of being endless changes somehow.

I don't know what that means. This shouldn't come off as surprise to you for the simple reason that I never said such a thing. I never said that adding an element to an infinite set means changing its property of being endless. What I said is that it increases the number of elements it has (the set remains endless.)

The best I can do is to imagine adding something to a collection of things that's already infinite and asking myself: does this change the property of being endless. And the answer seems to be: no, it's still endless. If you say, it makes it more endless, then I have to ask about that in the same vein: what am I imagining when I imagine something being more endless than something else that's endless?

I don't know what it means to say that a set is "more endless" than it was before. Sets are either endless or they are not. There are no degrees fo endlessness.

What happens to an infinite set when you add a new element to it is the same thing that happens to finite sets: you make them bigger. They have everything they had before + this new thing.

The problem is that infinite sets are endless so you cannot simply list of all their members (something you can do with finite sets) which makes it all too easy to pretend that adding a new element to an infinite set does not change that set.

So you agree that endlessness is a property of sets, one that sets either have or they don't. Do you agree that the word "infinite" is just another word for endlessness?

When you say you make an infinite set bigger by adding a new member to it, would you describe this as "more infinite"?

When you say \(\infty\) + 1 > \(\infty\), you obviously don't mean "more endless" since you just agreed that endlessness is a property that a set either has or doesn't have--no degrees--and I'll await your reply to see what you say about "more infinite"--but could \(\infty\) above be replaced with [the size of an infinite set] + 1 > [the size of an infinite set]?

I can see how it might seem intuitive that adding a member to an infinite set would increase the overall number of members in that set, but because the number of members of the set is directly tied into its property of being infinite/endless (in fact, defining it as such), that for me brings into question what things we can say about its size (number of members) and what things we can't say.

The point of the parallel lines thought experiment I brought up was to show how when you remove a subset of points from one of the lines, making it appear to have changed, rearranging the remaining points can show that nothing has really changed in the line at all. It reacquires its prior state exactly. I know we had a disagreement about whether it was really the same or there was a persistent change, but I explained that it's not about which points are which but that the structure remains the same. I used the building analogy to solidify this point (you didn't reply to that which lead me to wonder whether you read or not, so let me know if this doesn't ring a bell). The overall point is that if we want to say that something about an infinite set changes when we add or remove elements from it--whether that's its size, length, weight, density, whatever--I'm expecting to be able to detect a difference between the set's before and after states. But that's the problem with infinity. Infinite sets turn out to be very resistant to change, at least in terms of their quantity. Assuming uniformity of all its members, the things in it can be rearranged and shuffled around such as to assume exactly the same structure after adding or removing elements as before. And for this reason, I don't find it as easy as you to simply say the size of the set increases/decreases by adding/removing members.

[Magnus Anderson]

I take it you think that because the infinite sum of 0.9 + 0.09 + 0.009 + ... never attains 1 (or just never completes), it doesn't even attain the status of being a number. Is that right?

Not really. I didn't say it's not a number. I said that it does not evaluate to a finite number.

\(0.9 + 0.09 + 0.009 + \cdots\) is an infinite sum that does not evaluate to a finite number. This is because there is no finite number it attains.

Infinity obviously falls outside the range of the number line (what I mean by "beyond numbers") and so it does not fall on the number line, and therefore isn't a number.

That's what I call being fooled by the appearances. The number line represents the set of real numbers and the set of real numbers is not the set of all numbers that can be imagined -- it's merely a small subset of it.

Have you heard of the hyperreal number line? That's a different kind of number line. It not only includes infinity, it also includes different degrees of infinite and infinitesimal quantities.

Because arithmetic deals with numbers and only numbers (which are, by definition, finite).

They aren't finite by definition. I'm not aware of any such definition. It's just that we are mostly familiar with finite numbers (since they are of the greater relevance to us.)

Quantities have to do with more/less. Anything that can be said to be less than (or more than) something else is a quantity.

You're calling the assertion that arithmetic only deals with finite numbers "popular opinion", huh? I'd call it self-evident, pretty close to axiomatic--that's just what arithmetic is.

You can ask me to justify my claims ad nausium, but we're getting pretty close to elementary principles here, which is to say rock bottom, end of the line. Soon it begins to be like asking: why does 1 + 1 = 2? Little children do this. The incessant chain of never-ending whys. Louis C K has a good bit on this:

Every "Why?" question can be answered provided that you're interested in doing so. If you aren't, you don't need to justify yourself. You can simply leave the discussion.

The point is to examine each other's assumptions and see who's at the wrong side. But again -- you don't have to do this if you don't want to.

So you agree that endlessness is a property of sets, one that sets either have or they don't. Do you agree that the word "infinite" is just another word for endlessness?

Yes.

When you say you make an infinite set bigger by adding a new member to it, would you describe this as "more infinite"?

That's not how I would call it.

The point of the parallel lines thought experiment I brought up was to show how when you remove a subset of points from one of the lines, making it appear to have changed, rearranging the remaining points can show that nothing has really changed in the line at all.

But rearranging the remaining points does not show that the line didn't change. It merely makes it look like it didn't change. It's a magic trick.

I used the building analogy to solidify this point (you didn't reply to that which lead me to wonder whether you read or not, so let me know if this doesn't ring a bell).

I did but I didn't get to respond to it.

The overall point is that if we want to say that something about an infinite set changes when we add or remove elements from it--whether that's its size, length, weight, density, whatever--I'm expecting to be able to detect a difference between the set's before and after states.

When we say that you should imagine yourself standing in front of an infinite line of green apples, what we're doing is creating an imaginary universe in which you participate.

The creators of this universe are its gods (which is really only me) and you are one of its participants. Since I'm a god, I know what is true. And since you're not, you may or may not know what is true.

One of the gods of this universe proclaims "There is a man called Gib standing in front of an infinite line of immovable and indestructible green apples. One day, the gods gift him with a red apple and he decides to place it in front of the green ones." The question here is whether it follows from this description, i.e. whether the gods are making a statement, that the line is longer than it was before.

The gods may continue and say "This man called Gib, as well as every other man that shall ever exist, will never be able to know whether any two infinite lines are equal in length or not." That does not mean that the gods didn't say that the line became longer after addition, and also, it does not contradict that statement because what is and what can be known are two different things.

The only thing we want to do here is understand the kind of universe that I've just described.

[phyllo]

Not really. I didn't say it's not a number. I said that it does not evaluate to a finite number.

Well it's not an infinite number. Having an infinite number of digits in a base-10 representation does not make a number infinite.

"Infinite" is being used to refer to the number of digits in the representation of a number, the number of terms in a series expansion, the index of a term in a series, the number of elements in a set, the index of a element in a set, the sum of a series and the value/size of a number.

It's no wonder that this thread is a confusing mess of mixed up concepts, ideas and results.

[Magnus Anderson]

Well it's not an infinite number.

It's certainly less than \(1 + 1 + 1 + \cdots\). But it's not finite either.

Having an infinite number of digits in a base-10 representation does not make a number infinite.

Yes.

[Ecmandu]

Magnus, if you add a 1 at the beginning or end, you are adding a new dimension, not increasing the number...

You know, like IP addresses!

234.564.108.22

Those dots stand for different dimensions..

[Magnus Anderson]

You're incorrect that I'm "relabelling" the elements of set \(A\). I'm adding \(1\) to each of their elements and appending the result to the \(0\) that I first append to \(B'\).

Yes, you're adding \(1\) to each of their elements, and by doing so, you're relabelling them.

I don't give a shit what anyone thinks of me

I give zero shits.

If you have to say it, you do.

Maybe you should say it one more time?
Just to be sure?

I'm willing to wait that long for you, Magnus.

You are officially insane.

You cannot logically declare an infinite series of zeroes entirely bounded by a decimal point and a \(1\) because entirely bounded contradicts the boundless quantity of zeroes in between. Simple. Logic.

An infinite sequence such as \((1, 3, 5, \dotso, 6, 4, 2)\) is not a logical contradiction even though its bounded from two sides. This is because the ellipsis is not telling you that the sequence has no beginning (first element) and no end (last element.) It's merely telling you that the number of elements has no end.

I'd be more than happy to continue this discussion with you, but first, you'd have to stop being so hysterical.

[Ecmandu]

An infinite sequence such as \((1, 3, 5, \dotso, 6, 4, 2)\) is not a logical contradiction even though its bounded from two sides. This is because the ellipsis is not telling you that the sequence has no beginning (first element) and no end (last element.) It's merely telling you that the number of elements has no end.

I'd be more than happy to continue this discussion with you, but first, you'd have to stop being so hysterical.

Yes, there is a logical contradiction. 1,3,5 ... (say infinite 1’s) is a rational number - no problem there!

Tacking 6,4,2 at the end is not how math works.

[phoneutria]

there's an infinite amount of divisibles between any two numbers yo
you can do (0...1) and that is an infinite set
that is how math works

[phoneutria]

in the case of (1,3,5,…,6,4,2) both ends of the sequence go toward infinite counting by twos, one ascending and one descending, nerd.

[gib]

I take it you think that because the infinite sum of 0.9 + 0.09 + 0.009 + ... never attains 1 (or just never completes), it doesn't even attain the status of being a number. Is that right?

Not really. I didn't say it's not a number. I said that it does not evaluate to a finite number.

You agreed that \(0.\dot9 \neq \infty\). And it's not a finite number. What's left?

Have you heard of the hyperreal number line? That's a different kind of number line. It not only includes infinity, it also includes different degrees of infinite and infinitesimal quantities.

I'm somewhat familiar with the concept. I thought maybe that's what you were getting at with your diagram:

\(\bullet \bullet \bullet \cdots + \bullet \bullet \bullet \cdots\)

^ Sort of a whole new set of numbers beyond the real numbers ("beyond infinity" so to speak), but we didn't end up going there.

If this is what you're talking about, then I think the inequality:

\(\infty\) + 1 > \(\infty\)

is simply poorly written. I would write it:

R + 1 > R

where R is a hyperreal number.

\(\infty\), as I said, is, at least to me, not a number but a description of a set: it's infinitely large. It's a property of the real numbers (and I would guess, of the hyperreals as well). But if you want to do arithmetic on "infinitely large numbers", you could *probably* do it on hyperreals where such numbers are denoted R (or some variation). I wouldn't use \(\infty\) 'cause that's just confusing.

To me, hyperreals are just "numbers in a different universe" so to speak, so I don't *think* there would be any problem doing arithmetic with them. This is sorta what I was getting at when I talked about "reaching infinity" as being like reaching the edge of the universe, going beyond, and in a manner of speaking, entering a different universe (this was pages and pages and pages ago). This is where I introduced the idea of speaking of infinity as a unit and never speaking of the infinite quantity it represents again. It's because, once you enter this different universe, it's as if the entities you left behind in the previous universe no longer exist relative to the new universe. This is just a fancy way of saying the reals are an infinite distance away from the hyperreals and therefore might as well be treated like they don't exist. This is sort of why we call the normal numbers "real" and the ones beyond infinity "hyperreal" (I'm guessing mathematicians probably learned their lesson from calling \(\sqrt{-1}\) an "imaginary" number).

So far in this discussion, I haven't granted the reality/validity of the hyperreals (and my research on the internet tells me not all mathematicians do either), but I will grant that if you really want to develop a mathematical system based on numbers "greater than infinity" you could probably do so without running into too many internal inconsistencies. But its relation to the standard mathematical system based on real number would have to be thought through very carefully. For example, does it make sense to mix real numbers and hyperreals in arithmetic? Or if you want to quantify an infinite set, which hyperreal number do you choose? And do hyperreals have their own infinities (i.e. an infinite distance on the hyperreal number line)? Is that what 2 x R would equal (R being a hyperreal number)? Or would it, like real numbers, just be another hyperreal number on the same number line that's twice the distance as R is from the zero point on that number line?

I'm still not gonna grant that hyperreal numbers actually exist. I don't believe that if the edge of the universe is an infinite distance away, the possibility of an abstract mathematical system of hyperreal numbers proves that you can add one more unit of distance to that infinite distance and step into another universe that's a hyperreal distance away. Such a mathematical system would just be an abstract idea. But if we run with that idea in this discussion, it might be interesting to see what comes out of it. The first thing I would want to do is to agree on some ground rules--what can and what can't be said about hyperreals--or at least, lay out my assumptions.

Every "Why?" question can be answered provided that you're interested in doing so. If you aren't, you don't need to justify yourself. You can simply leave the discussion.

Yes, you can invent new answers. Just like Louis CK (jokingly) did when he said: "Well, because things that are not can't be!" <-- In his bit, this started from his daughter asking why she couldn't go out and play in the rain. Do you really think before he said "no" he thought "things that are not can't be... yada yada yada... therefore, my daughter can't play in the rain." The point is, you'll be doing that ad nauseum if your opponent's motive is to unearth the grounds from under your claims. He'll just keep asking why until you're not able to invent a satisfactory answer.

The point is to examine each other's assumptions and see who's at the wrong side. But again -- you don't have to do this if you don't want to.

That's a very noble motive, but in my experience, that's very rarely what's driving people in a heated debate. Most of the time, each party wants to destroy the other's argument--it isn't to share in the goal of finding the truth--which is why a lot of people will simply relentlessly demand justification upon justification upon justification... until their opponent is either exhausted or tongue tied or mistakenly says something that contradicts something else they said earlier (which is far more likely to happen if you're pressed to invent justifications on the spot). It's more of a game, a competition, than an honest and cooperative pursuit for the truth.

When you say you make an infinite set bigger by adding a new member to it, would you describe this as "more infinite"?

That's not how I would call it.

Right, you would simply say "it's bigger" or "contains more objects", etc. So you would reserve the word "infinite" to mean "endless" and you would agree it describes a property of a set with an infinite number of elements, and that it wouldn't change by adding or removing elements from that set. But you do think the number of elements in the set increases by adding 1 to it. Would this be represent as R + 1 where R is a hyperreal number?

But rearranging the remaining points does not show that the line didn't change. It merely makes it look like it didn't change. It's a magic trick.

Sure, but only in the sense that rearranging the bricks in a building doesn't mean the building hasn't changed. But that's irrelevant to comparing it to another building and asking: do they still contain the same number of bricks?

The overall point is that if we want to say that something about an infinite set changes when we add or remove elements from it--whether that's its size, length, weight, density, whatever--I'm expecting to be able to detect a difference between the set's before and after states.

When we say that you should imagine yourself standing in front of an infinite line of green apples, what we're doing is creating an imaginary universe in which you participate.

The creators of this universe are its gods (which is really only me) and you are one of its participants. Since I'm a god, I know what is true. And since you're not, you may or may not know what is true.

One of the gods of this universe proclaims "There is a man called Gib standing in front of an infinite line of immovable and indestructible green apples. One day, the gods gift him with a red apple and he decides to place it in front of the green ones." The question here is whether it follows from this description, i.e. whether the gods are making a statement, that the line is longer than it was before.

The gods may continue and say "This man called Gib, as well as every other man that shall ever exist, will never be able to know whether any two infinite lines are equal in length or not." That does not mean that the gods didn't say that the line became longer after addition, and also, it does not contradict that statement because what is and what can be known are two different things.

The only thing we want to do here is understand the kind of universe that I've just described.

You could think of it like that, sure. But be careful with this: "Since I'm a god, I know what is true." <-- This might hold in terms of setting up the parameters that define the thought experiment, but it doesn't mean you get to impose whatever conclusion you want. You can't just say: "I'm a god and this is my universe, therefore if I say the number of apples increases when gib adds one, then the number of apples increases. QED!" Obviously, in that case, you could say whatever you want, and I can say: "That's just an imaginary scenario that has nothing to do with reality."

But what I mean when I say "I'm expecting to be able to detect a difference..." is not that I in the scenario expect to detect a difference (indeed, I couldn't know), but I the thinker, the one imagining the scenario (I'm a fellow god too ), expect to detect a difference. I'm saying I need a way of conceptualizing the infinite set before the change as different from after the change. You're saying: just imagine it as having one more apple. But then I say: that's hard to do when the line is infinite. And I bring up the example of the line whose points were removed and the remaining points shift to take their place as an example. Even if we say the line now has fewer points, I can't conceptualize that. I can't tell (visually or abstractly) where there's a difference between the one line and the other.

[Silhouette]

You're incorrect that I'm "relabelling" the elements of set \(A\). I'm adding \(1\) to each of their elements and appending the result to the \(0\) that I first append to \(B'\).

Yes, you're adding \(1\) to each of their elements, and by doing so, you're relabelling them.

Different quantities have different labels - does that mean you're "relabelling" the same quantity? Of course not! Does it mean that a different label is the only difference? No!
Again!! The superficial consumes you and your thought, to yield superficial conclusions with no bearing on the reality that they represent.

I don't give a shit what anyone thinks of me

I give zero shits.

If you have to say it, you do.

Maybe you should say it one more time?
Just to be sure?

Anything for you, Magnus!

The quantity of shits that I give is the difference between \(1\) and \(0.\dot9\)

No wonder the more I explain the less you understand - you think that reiteration to help you understand makes something less true and that it must be only emotionally driven.
The superficial has consumed you!

I'm willing to wait that long for you, Magnus.

You are officially insane.

Sure, whatever you need buddy. I can be as sane or insane as you need me to be in your mind, and the logic of my arguments will stand.
For the 4th time, your judgments of me make zero difference to the undeniability of the reasoning that just happens to being delivered by me - but it could be anyone and it would be just as true.
The superficial is at your core, and you at the core of it.

You cannot logically declare an infinite series of zeroes entirely bounded by a decimal point and a \(1\) because entirely bounded contradicts the boundless quantity of zeroes in between. Simple. Logic.

An infinite sequence such as \((1, 3, 5, \dotso, 6, 4, 2)\) is not a logical contradiction even though its bounded from two sides. This is because the ellipsis is not telling you that the sequence has no beginning (first element) and no end (last element.) It's merely telling you that the number of elements has no end.

I'd be more than happy to continue this discussion with you, but first, you'd have to stop being so hysterical.

Stop caring about your judgments of me - they don't affect the irrefutable arguments laid before you.
The only person who matters here is you, and your learning to accept reason.

Again you're putting words in my mouth that I've not said, to make a straw man for you to attack.

The closest thing to what you think I've said is that the set as a whole has a beginning (first element) and an end (last element), and so do the 3 specified elements at the start and those at the end.
Therefore since the number of elements has no end (as you clarified is meant by the ellipsis), it's either impossible for the specified last 3 ending elements to ever come at the "end" of the "never-ending" ellipsis, and/or impossible for the ellipsis to ever even start if you're specifiying an end to the ellipsis for the last 3 ending elements to be able to ever be arrived at and be able to continue from what's in the ellipsis, and/or the ellipsis means a continuation of both progressions from both sides at once "inward", in which case the endless ellipsis has all ends defined and is disappearing up its own arse with no "middle point" that can ever be reached to connect each side - making it two endless sets bolted together by their lack of end, which is a contradiction.

You pick which impossible interpretation you want and it'll still be contradictory.

I'll just wait for the penny to drop for you over here, like an insane person

[Magnus Anderson]

there's an infinite amount of divisibles between any two numbers yo
you can do (0...1) and that is an infinite set
that is how math works

in the case of (1,3,5,…,6,4,2) both ends of the sequence go toward infinite counting by twos, one ascending and one descending, nerd.

True and true.

[Magnus Anderson]

Again you're putting words in my mouth that I've not said, to make a straw man for you to attack.

The closest thing to what you think I've said is that the set as a whole has a beginning (first element) and an end (last element), and so do the 3 specified elements at the start and those at the end.
Therefore since the number of elements has no end (as you clarified is meant by the ellipsis), it's either impossible for the specified last 3 ending elements to ever come at the "end" of the "never-ending" ellipsis, and/or impossible for the ellipsis to ever even start if you're specifiying an end to the ellipsis for the last 3 ending elements to be able to ever be arrived at and be able to continue from what's in the ellipsis, and/or the ellipsis means a continuation of both progressions from both sides at once "inward", in which case the endless ellipsis has all ends defined and is disappearing up its own arse with no "middle point" that can ever be reached to connect each side - making it two endless sets bolted together by their lack of end, which is a contradiction.

You pick which impossible interpretation you want and it'll still be contradictory.

Basically, what you're saying is that we cannot reach the last position of the sequence \((1, 3, 5, \dotso, 6, 4, 2)\) by starting at the first position (occupied by \(1\)) and then moving a finite number of positions to the right. That's correct.

If you're saying that this implies the sequence has no end, then I would strongly disagree. It merely means you cannot reach the last position the way you're trying to reach it (i.e. by making a finite number of steps.)

[Silhouette]

Basically, what you're saying is that we cannot reach the last position of the sequence \((1, 3, 5, \dotso, 6, 4, 2)\) by starting at the first position (occupied by \(1\)) and then moving a finite number of positions to the right. That's correct.

I know.

If you're saying that this implies the sequence has no end, then I would strongly disagree. It merely means you cannot reach the last position the way you're trying to reach it (i.e. by making a finite number of steps.)

Can't reach it with an infinite number of steps either. There's always infinitely more when it comes to infinity...

Believe it or not, infinity literally means no end - clue is in the word.
One might even say this is only the infinity'th time I've pointed that out..., which I guess is why we'll never get there

[Magnus Anderson]

If you say "There is an infinite line of apples in front Joe" that does not mean that you cannot add "One day Joe ate them all". The two statements aren't contradicting each other. The second statement is not implicitly stating that there was no infinite number of apples in the first place.

[Silhouette]

If you say "There is an infinite line of apples in front Joe" that does not mean that you cannot add "One day Joe ate them all". The two statements aren't contradicting each other. The second statement is not implicitly stating that there was no infinite number of apples in the first place.

If "One day Joe ate them all", and "There is an infinite line of apples", there would be always be more left to eat if the line was infinite. A contradiction.

There is no final bound to endlessness that allows the completion of "all".
The series is only "complete" in that it is consistent and predictable in its infinite progression.

there's an infinite amount of divisibles between any two numbers yo
you can do (0...1) and that is an infinite set
that is how math works

Math works by defining from the finite down toward this infinite amount of divisibles.

If it tried to work starting from an infinite set between two finite integers it would never get off the ground, never mind getting from one finite bound to the other - Zeno style.
This is only expected, due to the contradiction in the premises of entirely bounding a set (with finites) and calling that set boundless (infinite).

You solve Zeno by defining from finites (hence the tautology in terms) instead of defining from infinites (a contradiction in terms).
That way you eventually get to the problems of infinity anyway, but you get there via a structure that you wouldn't have if you worked from the problems of infinity.

This has been the point I stated right at the start of my contributions to this thread, but due to others getting bogged down in various details that are confusing them, discussion about my whole point never really began.
Arising from a fundamental tenet of my own original philosophy, Experientialism, utility (e.g. finites) defies truth (e.g. infinites) at a basic level - i.e. experience is continuous: there are no gaps of nothingness in it to genuinely separate "discrete experiences" from one another. Yet without treating Continuous Experience as "discrete experiences" we can't do anything useful with it, or even say anything at all. You need to artificially dissect continuity in order to garner knowledge from it "about it", even though any subsequent analysis resulting from this inevitably leads to problems such as those presented by (0...1) anyway a posteriori - as is consistent with the self-evident experience as continuous "before" knowledge a priori that you started with in the first place.

[phoneutria]

Math works by defining from the finite down toward this infinite amount of divisibles.

If it tried to work starting from an infinite set between two finite integers it would never get off the ground, never mind getting from one finite bound to the other - Zeno style.
This is only expected, due to the contradiction in the premises of entirely bounding a set (with finites) and calling that set boundless (infinite).

You solve Zeno by defining from finites (hence the tautology in terms) instead of defining from infinites (a contradiction in terms).
That way you eventually get to the problems of infinity anyway, but you get there via a structure that you wouldn't have if you worked from the problems of infinity.

I solve Zeno by taking a fucking step forward.
It's not an infinite space. It's an infinitely divisible space.
Likewise, I can divide an apple infinitely, and that does not mean I have infinite apples.
I can spend an infinity of time eating it by always taking a bite that is half of what's in my hand, but I still only ate (slightly less than) half of an apple.

To play with Zeno's idea in an infinite space, where do you set the middle point?

This has been the point I stated right at the start of my contributions to this thread, but due to others getting bogged down in various details that are confusing them, discussion about my whole point never really began.
Arising from a fundamental tenet of my own original philosophy, Experientialism, utility (e.g. finites) defies truth (e.g. infinites) at a basic level - i.e. experience is continuous: there are no gaps of nothingness in it to genuinely separate "discrete experiences" from one another. Yet without treating Continuous Experience as "discrete experiences" we can't do anything useful with it, or even say anything at all. You need to artificially dissect continuity in order to garner knowledge from it "about it", even though any subsequent analysis resulting from this inevitably leads to problems such as those presented by (0...1) anyway a posteriori - as is consistent with the self-evident experience as continuous "before" knowledge a priori that you started with in the first place.

seems obvious

[MagsJ]

I rarely feel cornered, and have not been so far during this thread so your observation skills have unfortunately been mistaken.
If you felt as though I was, then regretably you have misjudged my comprehensiveness as corneredness though you have correctly identified my frustration with my feeling of houndedness by the tedious repetition of falsity.

I see.. I’d say, that infinite objects can only exist in an infinite space (universe), but concepts are not bound by these physical limitations.. obviously. So adding more apples to a set, could not then make that set become infinite.. it would just make that set larger.

The term infinite is therefore self-defining, so either something is infinite or it is not.

[Silhouette]

I solve Zeno by taking a fucking step forward.

You demonstrate grounds to refute Zeno by taking a step forward (fucking or otherwise), you don't solve it.
You solve it by identifying the flaws in the premises, correcting them and ending up with Experientialism.

It's not an infinite space. It's an infinitely divisible space.

To play with Zeno's idea in an infinite space, where do you set the middle point?

Infinitely divisible space.... into what? Infinitesimals? And yet there's always something smaller tending towards no other value than zero...
Any non-zero value infinitely many times simultaneously satisfies never getting beyond infinitesimal space because the infinitesimal is so small, and getting to infinite space because the infinite is so large. It's undefined, and treating it with any specificity violates its infinitude - so whilst being both infinitesimal and infinite... it's neither. Being "undefined", as the infinite is, is clearly problematic, no?
And yet, if we take the limit of infinite division that is "zero", even infinite zeroes would not seem to suggest any space at all.

This is why knowledge is defeated from its very start when it tries to begin with the infinite - because knowledge, by definition, needs definition: it needs finitude.

There is no middle point in an infinite space - the same reason why there's no centre of the universe even though everything's moving away from everything else. You need relativity to arbitrarily choose a point of reference to measure something against finitely, and even then only get an answer specific to that point of reference. The same thing applies to defining 1+1=2, you have to establish an empty set first: pure conceptual boundary. Only then can you define the successors, equalities and inequalities necessary to define 2. The problem with the infinity in between 1 and 2 (same as the problem between 0 and 1) only crops up after that once one at least thinks they've gotten off the ground.

Likewise, I can divide an apple infinitely, and that does not mean I have infinite apples.
I can spend an infinity of time eating it by always taking a bite that is half of what's in my hand, but I still only ate (slightly less than) half of an apple.

Clearly dividing an apple infinitely does not yield infinite apples.
Eating half an apple, then half of that and half of that, which I believe is what you're saying, leaves partial products of slightly more than nothing at all and the limit of the process approaches zero apple.

This has been the point I stated right at the start of my contributions to this thread, but due to others getting bogged down in various details that are confusing them, discussion about my whole point never really began.
Arising from a fundamental tenet of my own original philosophy, Experientialism, utility (e.g. finites) defies truth (e.g. infinites) at a basic level - i.e. experience is continuous: there are no gaps of nothingness in it to genuinely separate "discrete experiences" from one another. Yet without treating Continuous Experience as "discrete experiences" we can't do anything useful with it, or even say anything at all. You need to artificially dissect continuity in order to garner knowledge from it "about it", even though any subsequent analysis resulting from this inevitably leads to problems such as those presented by (0...1) anyway a posteriori - as is consistent with the self-evident experience as continuous "before" knowledge a priori that you started with in the first place.

seems obvious

Great! We're on the same page.
And yet, it's funny to call something obvious when nobody's ever thought to suggest it: that utility and truth are at odds with one another.
Everyone has always regarded them as complementary - and no wonder: isn't it odd that "what works" is based on a deficit of truth?

I see.. I’d say, that infinite objects can only exist in an infinite space (universe), but concepts are not bound by these physical limitations.. obviously. So adding more apples to a set, could not then make that set become infinite.. it would just make that set larger.

The term infinite is therefore self-defining, so either something is infinite or it is not.

Yes, if you want to make the thought experimental physical, you'd need an infinite space (universe), and yes, concepts are not bound by these physical limitations.

Adding more apples to a finite set would make that set larger, no issue there at all.
The issue is that adding apples to an infinite set of apples doesn't make "infinity bigger" because expanding the bounds of something to make it bigger only applies to the bounded and not the boundless.

The term infinite defies definition by the definition of "definition" and of "finite". Either something is infinite or it is not - of course. Infinite is only "definable" insofar as we can easily define finite... and then saying "not that". This says what you don't have and not what you do have. The analogy I used is that this "defines" what's in a hole by defining the boundaries of the hole and what's outside of it (i.e. it doesn't define what's inside the hole at all).

[phoneutria]

Clearly dividing an apple infinitely does not yield infinite apples.
Eating half an apple, then half of that and half of that, which I believe is what you're saying, leaves partial products of slightly more than nothing at all and the limit of the process approaches zero apple.

approaches half apple

Great! We're on the same page.
And yet, it's funny to call something obvious when nobody's ever thought to suggest it: that utility and truth are at odds with one another.
Everyone has always regarded them as complementary - and no wonder: isn't it odd that "what works" is based on a deficit of truth?

it's not an original thought
all of scientific research operates with this in mind
utility is a compromise

[Ecmandu]

Yeah! I get to play devils advocate to Magnus for a moment!

Magnus writes a number like:

2,4,6, 1... (1 repeating)

Now!

1,1,1,1.... (repeating) is possibly 8 smaller in VALUE, but not correspondence!

For example:

3.000...

Is smaller in VALUE than PI.

Correspondence however is the same.