Is 1 = 0.999... ? Really?

It won’t take eternity. You don’t need eternity in order to come to conclusion that every element in (A) is also present in (B) and vice versa. That wasn’t my point.

My point was that in order to determine whether any two infinite sets are equal in size or not, you have to know how each member of one set is represented in the other set. This can be as simple as “The way elements appear in one set is the way they appear in other sets”. If we said that this applies to (A = {1, 2, 3, …}) and (B = {1, 2, 3, …}) then we would be correct to say that the two sets are equal in size. But if a different rule applied (e.g. every number in (B) is represented by an odd number in (A) and every even number of (A) is represented by a non-numerical symbol in (B)) then a different conclusion would follow.

Note that (\bullet \bullet \bullet \bullet \cdots) is not a set, it is a sequence. Remember that unlike sets, sequences allow repetitions. Your confusion is created precisely by the fact that every element in the sequence (\bullet \bullet \bullet \bullet \cdots) is identical to every other. There is a number of ways to resolve this confusion. One way is to represent this sequence as a set. So instead of (\bullet \bullet \bullet \bullet \cdots), we’d have (L = {P_1, P_2, P_3, P_4, \dotso}) where “P” stands for point. Let’s say that the way elements appear in one set is the way they appear in all other sets. Take every odd point out and you get (L’ = {G_1, P_2, G_2, P_4, \dotso}) where “G” stands for gap. Remove the gaps and you get (L’’ = {P_2, P_4, P_6, \dotso}). Now, compare (L) to (L’‘). Do they have the same number of elements? Of course not. (L) has all of the elements that (L’') does plus some more.

Magnus,

in quoting James “that it’s to small to measure”

What’s being stated here is that it’s too small to explain. Explanations are measurements!

Yet, here he is, “explaining” it

I’m coming at this argument from your logic, not my logic.

Your logic is that infinity can be quantified.

So I used your logic to prove that everything except the highest order of infinity (in magnitude) (based on your logic), is actually fractional and has to be, but NEVER additive (2 infinite lines) - instead, you have (2) 1/2 infinite lines which still equals 1 infinite line.

It’s arbitrary in the same way that (A = {1, 2, 3, \dotso}) is arbitrary. I could have chosen any other set. In the same way, I could have chosen any other mapping.

I said that every number in (B) is represented by a unique odd number in (A). Since (A) is more than the set of odd numbers, it follows that (A) has more elements than (B). In other words, it follows that (A) and (B) are not equal in size.

You’re starting with a different premise. You’re starting with the premise that the way elements appear in one set is the way they appear in other sets. Such a premise leads to a different conclusion – it leads to the conclusion that the two sets are equal in size.

Before removing the points, the two lines (A) and (B) can be represented using the following sets: (A = {P_1, P_2, P_3, P_4, \dotso}) and (B = {P_1, P_2, P_3, P_4, \dotso}) where (P) stands for “point”. Take the set (B) and remove every odd point. What do you get? There is a number of different ways you can represent the result.

The result (C) can be represented as (C = {P_1, P_2, P_3, P_4, \dotso}) but in such a case you have to make it clear that the way elements appear in set (B) is not the way they appear in (C). Therefore, you can’t say that, because every (P_n) occurs in both sets, that (B) and (C) are identical. You have to ask: how is (P_2) from (C) represented in (B)? Certainly not as (P_2) because no odd point from (B) is contained within (C). It is perhaps represented as (P_3) or (P_5) or some other odd point.

The result (C) can also be represented as (C = {P_1, P_3, P_5, P_7, \dotso}). In such a case, the rule that the way elements appear in one set is the way they appear in other sets applies. And for this reason, it’s easier to see that the sets (B) and (C) are not identical.

You can determine the size of infinite sets but before you can do that you have to know how elements from one set are represented in the other set.

That’s correct. Note that you can say that both A and B are made out of p1, p2, p3, etc. It doesn’t matter that these points belong to two different lines.

What matters is that the original line has points that the resulting line doesn’t: p1, p3, p5, p7, etc.

If you’re correct that a ray cannot be shorter or longer than another ray, then there must be a flaw in my argument.

Where’s the flaw?

Point the flaw in the argument instead of merely doubting it because the meaning of the word “shorter” confuses you.

How would they appear to be just as long? And what does it mean that their lengths are undefined?

The length of a line is simply the number of units (e.g. inches) that constitute it. If you have a line that is infinitely many inches long, and you take one inch out, you get a shorter line (since “shorter” means “of lesser length”.)

Gib said he wasn’t going to debate you in your own terms, I am.

It’s respectful to reply to this as well, rather than ignore it - were all trying to get at the truth here.

viewtopic.php?p=2755663#p2755663

I don’t have to respond to anyone unless I wish to do so. In general, I have little to no interest in responding to your posts.

You know you don’t have to respond.

But you did reveal yourself.

I tailor made an argument for you, using your own logic, to refute your logic.

It actually takes me time to do things like that, which is why I said it’s disrespectful not to reply.

I suppose you mean contradictory.

My objection is that if (0.\dot9) is a symbol representing the same infinite sum with the same infinite number of non-zero terms wherever it appears, it follows that (9 + 0.\dot9) is an infinite sum that has one non-zero term more than (0.\dot9). This is because it has all of the terms that (0.\dot9) does plus one more. And when you divide it by (10), the number of terms is preserved, so the resulting number, even though similar in appearance, isn’t really equal to (0.\dot9).

On the other hand, I have no idea what it means to say that at some end of the infinite sequence of 9’s there’s a spare 9. You’d have to clarify that.

What “ends” are equal? And what “ends” were previously not equal?

That’s precisely what you’re doing.

You can arbitrarily determine how elements of one set are represented in the other set. But once you do so, you can’t arbitrarily decide whether the two sets are equal in size or not.

I can say that odd numbers in (A = {1, 2, 3, \dotso}) are represented in the following way in (B = {1, 2, 3, \dotso}):

$$
1 \mapsto 1\
3 \mapsto 2\
5 \mapsto 3\
\cdots
$$

This means that (1) in (A) is represeted as (1) in (B), and (3) in (A) is represented as (2) in (B), and (5) in (A) is represented as (3) in (B) and so on.

Once you accept this, you can’t say that two sets are equal in size. They aren’t. Every member of (B) is a member of (A) but the reverse isn’t true.

But you are not listening, so you keep making one mistake after another and getting frustrated.

It’s not arbitrary. What you’re doing is arbitrary. You’re the one parting ways with logic.

You just contradicted yourself. You said you can’t define the word “infinite” and then you went on to define it by saying it means boundless.

Yes, the word “infinite” means “boundless” but it does not mean “boundless in every way one can think of”. It means “boundless in some ways” where “some ways” can be “one way”, “two ways”, “three ways” or “all ways”. Yes, it can mean “bounded in all ways” but not necessarily. Its exact meaning depends on the context.

When people speak of infinite sets, they are not talking about all-encompassing sets i.e. sets that contain literally everything there is, they are talking about sets that have an infinite/endless quantity of members. That’s why it’s not a contradiction in terms to speak of infinitely many things happening within a finite period of time.

Yes, you are still telling me what to think.

Yes, you are being silly. We may need an extensive proof to acquire a “deep” understanding of why 1+1=2 but I wasn’t talking about the “deep” understanding.

All in all, you’re being pathetic.

I didn’t say that James never used “size”. I said that I personally think that the word is hindering the understanding and that James spoke of “degrees” which I think helps the understanding.

Reference?

Magnus,

Silhouette is already familiar with this argument.

An infinity can never be fully expressed. BUT! Because this cosmos is infinite, when it tries to “be itself” it expresses itself as motion, as it has no end.

This means that infinity is a process. No amount of abstraction is going to express an infinity, other than motion itself TRYING to express it.

See, the thing is… we’re always approaching infinity, but we never get there. If we ever “got there”, the entire cosmos would be frozen forever… it wouldn’t exist.

You exist… which means that infinity is never “arrived at”… because it’s never “arrived at”, motion exists, also implying the finite (the whole numbers); our singular experiences.

I see.

But I don’t see why I shouldn’t speak of infinite size, length, amount, quantity, number, etc.

Imagine the reaction of scientists and philosophers around the globe if an argument of this sort actually could be demonstrated as being true for all of us…and not just true in Ecmandu’s head.

Magnus, I’m agreeing with you. Now you want to turn that around and pretend I was responding to something else? You said: “Given any two infinite sets, you cannot determine whether they are equal in size or not by looking at their elements.” Do you all of a sudden not agree that it would take an eternity to look at all their elements? If you’re saying there’s other ways to determine a one-to-one mapping between the two sets, I once again agree, but that’s a different statement.

And how do you determine the mapping between two sets? In the case of the infinite parallel lines, how do we determine the rule that tells us how to map points from one line to the other? Is it arbitrary? If it’s arbitrary, why can’t we say there is a one-to-one mapping between the points in line A and the points in line B after removing points in line B and moving the remaining points into the gaps?

The crux of your argument seems to be this: “Do they have the same number of elements? Of course not. (L) has all of the elements that (L’') does plus some more.”… which seems to be yet another version of your original argument, the one about how finite sets work. You remove elements from a finite set, and you get a smaller finite set. The “of course not” sounds like a justification from intuition. Your experience with finite sets leads you to intuit the same must be true of infinite sets.

I’ll give you credit for adding a layer of sophistication on top of it with your mapping argument, but I see the mapping as completely arbitrary. You can fix it with a bit of re-labeling. Take (L’’ = {P_2, P_4, P_6, \dotso}) and relabel the points (P_1, P_2, P_3, P_4). ← There! You have a one-to-one mapping again. Same points, different labels. Just like name tags. If you have a room full of people and they each have a name tag, they don’t suddenly become different people by swapping out their name tags. Don’t worry about not having enough points in (L’') for all the labels… it’s infinite.

On the other hand, if you think the identity of each point is intrinsic to the point itself (so “P2” for example is not just a label but essential to what point P2 is), then there is no way any two lines (or any two sets of points at all) are identical. The first point in line A must be labeled something different form the first point in line B, otherwise you’re saying they are the same point. But when I said the two lines are identical, I didn’t mean they share the same points, I meant there is no way of distinguishing which is which (short of where they are relative to each other), and certainly no way of determining whether one is shorter than the other.

And you don’t see the problem with this?

But couldn’t this be argued the other way around? If it’s arbitrary, why can’t you say every number in (A) is represented by a unique odd number in (B)? That way, B is the larger set. Are you saying the size of the set is relative to how you do the mapping, or that B is larger and smaller than A at the same time?

So far so good, Magnus. I agree. The labeling is arbitrary, but if we’re trying to preserve the mapping, we have to be clear about which points matches up with which other point. I also agree that there is more to how elements in a set appear than just their labeling. In line B, for example, when we removed every odd point, the points now appear with gaps between them, which makes the line appear different from how line A appears. This holds even if we relabel the points in line B to match the sequence in line A.

Fully agree again. Having a different labeling pattern does make it easier to see the difference. But are we once again forgetting that crucial step? You know the one I mean. Moving the points in line B to fill the gaps? Before taking that step, there is indeed a difference in how the points in line B appear compared to those in line A–there’s gaps between them–but once you fill the gaps, that difference goes away. The labeling doesn’t matter because it’s arbitrary. If the remaining points in line B after removing the odd points were labeled (P_2, P_4, P_6, P_8), then we could relabel them after they move to fill the gaps as (P_1, P_2, P_3, P_4) and there would no longer be a difference. You could even relabel the points in line A as (P_2, P_4, P_6, P_8) to make it look like line A was the one with fewer points.

That the logic of finite sets carries over to infinite sets.

They’re infinite! They remain just as infinite no matter how many points you remove! In what way would an infinite line look different after removing any number of points and moving the remaining points to fill the gap?!?! Infinite lines always look the same.

This is just another way of saying the same thing. This is for those who have trouble with the notion of two parallel infinite lines being “just as long”. If such a notion, to them, implies that they both start and end at the same spot (that is, length is necessarily finite) then we can say the lines don’t have a length, their length is “undefined”. This is sort of like the idea that because infinity is not a quantity, an infinite set has no quantity, its quantity is undefined because it is “beyond quantity”.

^ Take your pick–defined, undefined, largest quantity, beyond quantity–I don’t really care. To me, they mean the same thing. The point is, the length of both lines is the same. Both infinite, or both undefined.

Then I have to say, Magnus, line B is not shorter after removing points and filling the gaps with the remaining point because, well, there’s still an infinity of points. Number of units before removing points: (\infty) points. Number of units after removing points and shifting to fill the gaps: (\infty) points.

How do you determine the contents of the set (A)? Why (A = {1, 2, 3, \dots}) and not (A = {2, 4, 6, …})?

Due to logical consistency.

If you’re going to relabel the points, you’ll have to remember that the rule “The way elements appear in one set is the way they appear in other sets” no longer applies. (P_2) in (L) is no longer represented by (P_2) in (L’‘). Instead, it’s represented by (P_1). ({P_1, P_3, P_5, P_7, \dots}) remain unrepresented in set (L’').

If I call you Donald Trump, does that mean you’re Donald Trump? Of course not.

So re-labelling is merely a trick. There’s still no one-to-one mapping.

That’s unnecessary.

You can tell they are different thanks to logic. But if you’re not a fan of logic then . . .

Only people who don’t understand how logic works do.

We took the set (B = {P_1, P_2, P_3, P_4, \dotso}) and removed every odd point (we also removed the gaps.) The result is (C = {P_2, P_4, P_6, P_8, \dotso}). So no, no step was left out.

It does matter.

That doesn’t cut it. You have to show me the logical step that is mistaken.

You avoid doing this because you’d rather talk about your own independent arguments.

Infinite means endless. It does not mean “equal in size”.

They remain endless but their sizes change.

And yet one of those Wikipedia “proofs” claims that (9.\dot9 - 0.\dot9 = 9). If infinite sums aren’t quantities then you cannot subtract them.

Their lengths are undefined and at the same time equal (i.e. the difference between the two lengths is zero)?

Idiotic would be better, actually.
Pick one, they’re all entirely appropriate :-" even if you pretend only one (your one) is correct - oh wait, that’s what you’ve been doing this whole time - at least I’ll know what stupidity to expect.

Are you trying to ignore the connection between (\frac{9 + 0.\dot9}{10}) “having 1 more term” than (0.\dot9) and having “no idea what it means to say that at some end of the infinite sequence of 9’s there’s a spare 9”?
Or are you unable to see it? It’s one or the other.

(\frac{9 + 0.\dot9}{10}=0.\dot9)
So if it has “1 more term” than some other (0.\dot9) then where is it?
It’s not in front of the decimal place.
The 9s after the decimal place go on forever with no end. With these nonsense notions of (0.\dot{0}1) that you tried to conjure out of your ass before, you were pretending you could fix some term at the end of an endless sequence, so maybe you’re proposing this “spare 9” performs that logically contradictory feat? Nothing else works for you, so where are you going to retreat to now?
(0.\dot9) certainly has one-to-one correspondence with (0.\dot9)
Again, your logic works only for finitude - not infinitude.

In your visual display you start the purple (0.\dot9) after the green by shoving another yellow term in front of it that’s not even part of the purple set that you’ve pushed over anyway.
But even if the purple could be justified as starting after the green, it would only therefore be shorter if the ends of each (infinite) set “ended” at the same point.

But as I just explained above, your previous example tries to have two sets of (0.\dot9) “constructed in different ways” starting at the same point, but one “had an extra 9 added in at the beginning before all elements were shifted over” therefore each (infinite) set would have to “end” at different positions for one to be longer than the other.

Inconsistency. Arbitrary. Contradictory. Hypocritical of you to accuse these things of me.

Listening to you making one mistake after another and insisting you’re right is what’s getting me frustrated!

You literally created identical sets, i.e. bijective by definition, (A = {1, 2, 3, \dotso}) and (B = {1, 2, 3, \dotso}), and arbitrarily matched them with the appearance of one-to-one correspondence such that only the odd numbers in A are counted, and therefore concluding that A is a different size to its identical counterpart, B.

I’ll give you the benefit of the doubt and assume you meant the set, A (with only odd numbers) was being compared with B such that (A = {1, 3, 5, \dotso}) rather than (A = {1, 2, 3, \dotso}).
Even then the one-to-one correspondence is with the natural numbers, x in B with 2x-1 in the original A.
You’re wrong either way, and I’m right either way.

If you took just a minute or two out of your busy life of complacent assertion and read up on bijection, you would know that the function (f: R\mapsto{R}, f(x) = 2x - 1) is bijective.

You’re the one misunderstanding quantity and getting fooled by decimal notation such that you think that one-to-one correspondence between the first element in one set and the first element in another set is “wrong” because you put them in different columns in decimal notation.

This parts with logic on superficial arbitrary grounds.

Yes it’s so arbitrary of me to suggest one-to-one correspondence of first elements in a set rather than cooking up some pseudo-logic to match the first with the second just to give the illusion of different “size” - size of infinites no less! They all go on forever, some don’t go on “more forever” than others as I’ve said so many times and you’ve still not accepted.

No, you just conflated signifiers and with signifieds to think that was a contradiction.
Please tell me you think I made up that terminology so I can annihilate you all over again in yet another way.

All words obviously have definitions, but the things they refer to can defy definition - making the act definition questionable in the first place. “The defiance of definition” can suffice as a definition of a word in reference to some undefined aspect of existence. The signified has infinitude, but the signifier implies at least some finitude for it to be a word at all. There’s obvious problems with the truth in doing this, but undeniable utility in doing so - hence why people perform this questionable act in the first place.

Only in this way with these concessions can the word infinity have definition and even synonyms like boundless, which is only a “definition” in the same way that a tautology gives extra information (it doesn’t). There’s only an appearance of definition here (which again is your whole problem), and on top of that infinitude is an absence of finitude (definability) rather than a definable thing itself. Saying that which is infinite is provably definite is like asking someone to prove the nonexistence of absence - as well as being a logical contradiction.

There’s always so many things wrong with everything you say!!!

This is already how I ascertained you were thinking of infinite a week ago and here you are again thinking you’re telling me something I don’t already know/understand. Again.

I criticised the use of “infinite” in reference to sets when any aspect of their construction is a product of finitude. I mentioned how even the natural numbers have a finite starting point on the number line, never mind the line being finitely bounded in all other dimensions as well, being infinite in only one dimension in one direction. It’s finite in many more ways than it is infinite, yet it’s still called infinite because it’s infinite in at least one way. I also explained that even in other sets infinity is still in only one way, just with less and less finite constraints the more “types” of numbers you add. In other words, any size of “infinite sets” is determined by their relative lack of finite constraints and not any “different size of infinity”. It’s only if you could remove all finite constraints to “infinite” sets, that you’d get an entirely infinite set, which would mean “boundless in every way one can think of”. But this would require complete consistency - not your strong point.

Instead, you’re happy to refer to all sets that have infinity involved in them in with the exact same term, no matter how many more finite constraints that they also have “but just in a different context”. Stay as vague as you can to maintain only the amateur appearance of validity, right?

If you had a basket with several oranges and one apple in it, would you call it a basket of oranges? Obviously not, but people like you will flatly call any set “infinite” no matter how many more ways it had finitude involved in it.

I’m telling you TO think.
And I’m showing you how to do it logically.
If you do that then you’ll end up thinking what I think and what the professionals think, but for you to accept and understand it, you have to do it yourself (think logically).
But you can’t even seem to identify my explanations as explanations - simply asserting that they’re just demands and pretending you’re thereby unchallenged.
Clearly there’s something going wrong for you to object to thinking and doing it logically - but I can only lead a horse to water, I can’t make you drink think.

I’m being utterly serious, and your “Argument from Incredulity” (if I cannot imagine how this could be true, therefore it must be false) is pathetic.
I know you aren’t talking about a deep understanding!!!
Apparently you think a deep understanding of what we’re talking about is pathetic.

Enough said.

Principia Mathematica by Alfred North Whitehead and Bertrand Russell.

Building up the foundations of maths, they finally reach 1+1=2 by page 362 (in the 2nd addition at least, in the 1st addition it’s page 379).

Goddamnit, why is it so impossible for amateurs to believe that someone else might know things they don’t?
Dunning Kruger effect, obviously. But fuck me is it frustrating to have to deal with.

The yellow rectangle is equivalent to the blue rectangle. That’s why it’s beneath it. The yellow rectangle shows (9.0) which is what you get when you take all of the elements from the blue rectangle and sum them up ((10 \times 0.9 = 9)).

The purple rectangle is equivalent to the red rectangle. That’s why it’s beneath it. The purple rectangle shows (0.9 + 0.09 + 0.009 + \dotso) which is what you get when you take all of the infinite sums from the red rectangle and sum them up ((10 \times 0.0\dot9 = 0.\dot9)). The number of terms constituting the resulting sum shown in the purple rectangle is equal to the number of terms constituting infinite sums shown in the red rectangle. Every infinite sum in the red rectangle has one term less than the infinite sum shown in the green rectangle.

The purple rectangle does not start at the same place as the green rectangle for the simple reason that its equivalent, which is the red rectangle, does not equal (10) times the green rectangle but (10) times the green rectangle minus one term which is the first term (just as the image shows.)

But they aren’t identical. You are merely not listening.

That’s nonsense.

Are you one of those people who think that the symbol must look exactly like the symbolized in order for us to be able to say that the symbol represents the symbolized? and that otherwise, the claim that the symbol is representing the symbolized is at best a useful contradiction?

That’s utter nonsense.

It’s like saying the word “apple” doesn’t represent apples because it looks nothing like apples.

There is only one way that sets can be infinite.

(0.\dot01) or (0.000\dotso1) is merely a convenient representation of the infinite product ({\displaystyle \prod_{n=1}^{\infty} \frac{1}{10}} = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots ). The infinite product has no last term. The end is merely in the symbol representing the infinite product.

Let’s say you’re standing in front of an infinite line of green apples and one day you decide to add a red apple to it. Where did you add it? We can’t say exactly where because we don’t have enough information but we can say that it’s somewhere inside the line. It can be literally anywhere in the line. It can be at the beginning of the line or it can be 100 apples away from the beginning. One thing is sure: you didn’t put it next to the last apple in the line because there is no such thing as “the last apple in the line”. The point is that, just because you added a thing to an infinite line of things, it does not mean you placed it right after the last thing in the line.

What does it mean to say that an aspect of a construction of some set is a product of finitude?

The set of natural numbers does not have a starting point. There is no first element, member, number.

How is the set of natural numbers (not the number line) bounded in all other dimensions?

How can sets be infinite in more than one dimension and in more than one direction?

There is only one way that sets (including the set of natural numbers) can be finite or infinite.

What does it mean that sets have “finite constraints”?

A set is said to be entirely infinite if the number of its elements is endless. That’s what it means for a set to be entirely infinite. A set cannot be more or less infinite. It cannot be partially infinite. It’s either infinite or it is not.

Omfg :laughing: I know why you put it there!! #-o It’s just idiotic to use that as an excuse to shift over the separate purple set that doesn’t even include the yellow set that’s being used to superficially shift it.

This shit you’ve just wasted keyboard strokes on is not only unnecessary waste, it does nothing to excuse the fact that you’ve started a set “one to the right” just to say that it doesn’t start “one to the left” like the green one.

I know WHY you did it, it’s painfully obvious, but the result of playing around with superificial positionings doesn’t change the fact that green and purple have perfect bijection: the first terms match, the second do too etc. It’s just more sophistry by the uninitiated to fool the uninitiated.

Right. (A = {1, 2, 3, \dotso}\neq{B} = {1, 2, 3, \dotso})

What am I not listening to when you literally write out the exact same set twice? You saying they’re not the same even though they are?
So sorry for not believing you when you literally write out right in front of everyone the exact same set.

No it really very super isn’t! :laughing:

How can you not see that the definition of definition can’t apply to that which has no definition? And yet we do it anyway?

Presumably you think it’s nonsense because you lack the ability to see contradictions plain and simple right in front of you, even in your own “reasoning”?

Yes, it is utter nonsense to say the word apple doesn’t represent apples because it looks nothing like apples.
No I’m not “one of those people who think that” - that’s retarded. Who are these people you’ve been hanging around? I think they’ve had a bad effect on you, or at least your “reasoning” is explained by the company you’ve been keeping.

I’m saying the whole point of a word is that it isn’t what it represents - hence the whole distinction between signifier and signified that I brought up…

I’m saying the purpose of matching things that don’t otherwise match is for utility. As you say, it’s not true that the signifier has to match the signified. Hence the distinction between truth and utility.
I hesitate to introduce you to Experientialism, my own original philosophy, which notes this distinction as one of its most primary of tenets. I think it’ll just confuse you even more than you already are.

Exactly! Just as I mentioned last week.

Hey, maybe you have been listening…
I’m sure it’s just an accident, but keep it up even if it is.

The number of finite constraints around natural numbers differentiate them from integers even though there is only one way that each set can be infinite! Exactly.
Hence the negative integers bolted onto the zero that precedes the natural numbers doesn’t “double the size of the infinity”, it subtracts 1 finite constraint.

I have a dream… that we’re getting somewhere! Pinch me, Magnus.

Just so you know, you don’t need to state the n value when it has no bearing or mention on the infinite series.

A better notation might be (\prod_{n=1}^\infty\frac1{10_n}) or maybe even (\prod_{n=1}^\infty{10_n}^{-1}). N could even equal 0, it makes no difference.

Either way, (\lim_{n\to\infty}) of this infinite product is (0).
There is no other value that it approaches, because there’s always a next term that makes it 10 times less, and there is no “smallest” quantity at the “end” of an infinite series.
We can pretend there is, with Epsilon, ε, but we’re only pretending.

One apple at the “starting bound” of a “boundless” line of green apples doesn’t make it “more boundless”.
You could pretend the whole line moved towards you, or u took an apple-length step towards it to produce the same superficial appearance. It’s endless either way - superficials don’t change this. You added a quantity of 1 apple and nothing changed to the quality of endlessness - I’ve been saying this from the very beginning. The quality of having no quantity is not a quantity. Add it literally anywhere in the line, as you say - no difference. It would make a difference to a finite line, for sure. Adding a new first element to an infinite set just gives you an infinite set with a new finite bound - the finite “1st element” changed, shifting all successors down by 1 place infinitely… - no size change occurs. It would occur for a finite set, sure, but you can’t have a “longer” infinite endlessness even if you change a finite constraint to how it starts.

So you are telling us that someone took 362 pages just to prove that 1+1=2?

First, I don’t believe it. And then if they did they were definitely missing something upstairs.

I realize that appearances aren’t everything and no offense but compared to Magnus, you are the one who appears to be the amateur here suffering from Dunning-Kruger effect.

As he pointed out earlier, you have not shown any flaw in his argument. You just say he is wrong and then give your own narrative. If Whitehead and Russel argue like that, I can see why it took so long for them to do so little.