Is 1 = 0.999... ? Really?

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.

Here it is:

(\frac{9 + 0.\dot9}{10}=0.\dot9 + \underline{\text{the missing term}})

If you’re asking me about its value, I don’t know, I didn’t calculate it. Do you really think such is necessary in order to prove that there is indeed a missing term? I don’t think so.

What do you get when you take (1 + 1 + 1 + \cdots) and add one more term to it? You get ((1 + 1 + 1 + \cdots) + \underline{1}). The underlined is the added term. That’s where it is. In this particular case, it’s pretty easy to calculate the value of the added term because every term in the infinite sum (1 + 1 + 1 + \cdots) is equal to every other. This isn’t the case with (0.\dot9), so figuring out the value of a single term is not so straightforward. You’d have to find an equivalent infinite sum where every term is equal to every other and then calculate how many terms of that sum is equal to a single term of (0.\dot9).

What, you mean you just start enumerating the contents? I suppose you’re saying you enumerate the contents in both sets, and for every member you enumerate in the one set, you map it to a member in the other set, and that mapping counts as the enumerating of members in the other set. So you start with a1, and you map that to a member in the other set, and you call it b1. Then you do the same for a2, labeling it’s mapped counterpart b2. And so on.

In other words, there’s no rule per se–no order with which you must enumerate and map the members–the order/rule is derived based on the enumeration and mapping. So the rule becomes: a1 maps to b1, a2 to b2, etc. The rule is essentially: don’t mess up that order. Is this in the ball park?

So the mapping is based on identity. You’re saying you can’t just take b2 from line B and relabel it b1. You’re saying b2 is a specific point, not the same point b1 was (even though it may now occupy the position b1 once occupied), and therefore cannot be said to be identical to b1. It follows, therefore that the entire line cannot be said to be identical to what it was before (even though, by all accounts, it looks the same).

You must concede then that we cannot say line A is identical to line B even initially, for a1 is not the same point as b1, and therefore even though they look identical, they are not.

This misses the point, of course. The point of this thought exercise was to show that if we’re saying line B is the same as line A based on structure, then it must remain the same after removing points and shifting the remaining points because the structure of line B remains the same. ← This argument called them identical, not because every point in the one line was the same point in the other line, but because their structure is the same. It’s like if I said two buildings are the same because they have the exact same structure and appearance. I don’t mean to say every brick in the one build is a brick in the other building–that would make them a single building–I mean to say there is no way to distinguish them. You could even remove every odd brick in the one building and replace them with entirely new bricks. As long as the structure and appearance remained the same, we can still say they are identical.

Mapping has nothing to do with it. I could map brick a1 in building A to brick b1 in building B, and if b1 was one of the bricks removed, the mapping is broken. Whatever brick we bring in to replace it is not b1, and therefore requires a different label (say c1), and if we want to map a1 to c1, we’d have to concede that it is a different mapping from that between a1 and b1. But as you can see, this is completely irrelevant. We can still say the buildings are identical.

But you’re argument does revolve around identity, doesn’t it?

Then there must be a misunderstanding about what it means to say “the way elements appear in one set is the way they appear in other set”. I took it to mean, the way they appear visually. So when I visualize lines A and B, they appear the same. Then when I visualize them again after removing the odd points from line B, the obviously don’t appear the same. Finally, when I visualize them after shifting the points in line B to fill the gaps, they appear the same again. Or do you mean how they appear with the labels? Labeling a sequence of points P1, P2, P3, P4, P5 would certainly make them look different from an identical sequence of points labeled P2, P4, P6, P8, P10. But labels are accessories. It would be like saying building A and building B are no longer identical when we label them “building A” and “building B”. ← This misses the point of what we mean when we say two things are identical… unless, of course, you’re talking about identity. I’ll have to see from your response.

You did agree above that the labeling is arbitrary. You must have meant only initially (i.e. when we first enumerate the members of each set and determine their mapping). If you mean to say the mapping must remain consistent after that, I see your point.

That is the logical step that is mistaken. The problem here isn’t that I’m failing to point out the flaw, it’s that you don’t understand why it’s a flaw. And I’ve been trying, Magnus, believe me, but it’s not easy. There’s a huge difference between that and failing to even attempt to point out your flaws.

There’s only two options here. Either they’re equal in (infinite) size or the notion of size is meaningless with respect to infinite things (this is the “undefined” size view). In either case, they definitely appear just as long (which was what you were questioning).

a level of stupid i have not seen in a long time.jpg

H’oh, boy. Let’s see if I can use an analogy to explain what’s so deeply wrong with that statement. Two analogies. First, (0.\dot9) is like (0.\dot3) in that they both have infinite decimal expansions. But (0.\dot3) is not an infinite quantity. It’s just one third. (I chose (0.\dot3) because I can’t say (0.\dot9) is just 1 without you fighting me on it.) Mistaking the quantity represented with the representation is such a juvenile mistake. But just in case you mean something a little more sophisticated, I have a second analogy for you. By “infinite sum” I’m guessing you mean 0.9 + 0.09 + 0.009… In that case, both terms are the results of infinite sums. But then adding infinite sums is not the same as adding infinite quantities. So here’s the second analogy: 10 + 20 + 30 is a finite sum. It equals 60. What makes it finite is the number of terms. There are 3 of them. But you’re not adding 3. You don’t include the number of terms as one of the terms being added. You don’t add 10 + 20 + 30, then say: well, that’s 3 terms, gotta add 3 for 63. So if we had an infinite sum, like 0.9 + 0.09 + 0.009…, why would we add infinity? The sum of the terms is just 0.999… which is just 1, a finite number (or even if you disagree with that, it’s a number less than 1, but still definitely finite). With (9.\dot9 - 0.\dot9 = 9), you’re not subtracting infinity from infinity. You’re subtracting two very small finite numbers. You may have had to sum an infinite number of terms to get each one, but even that isn’t adding infinities. But regardless of whether infinity is a quantity or not, I’ve never come across a case of doing arithmetic with infinity and getting meaningful results.

You can say they’re both undefined, or both infinite. ← Those are the only two senses in which they are equal. You certainly can’t say they are unequal. On what basis could you say they are unequal? And which one is less/more than the other, and by how much?

Anywaaay, I want to get back to your mapping argument. Though I see a few flaws in it (not the least of which is that it misses the point), I think I understand what you’re trying to get at. You’re trying to get me to keep track of which point in the one line corresponds to which point in the other. You’re hoping to show me that if points that were side by side initially are no longer side by side in the end, then I might realize: wow, there is a difference after all. But even if I grant that, that makes no difference to the size of the lines (which was my point). Even if it’s no longer the same points side by side, the lines still appear the same, and appearances is all I need. To judge the length of something as equal to something else is all a matter of appearance. We say: the brick appears to be 10 inches long because it appears to be the same length as the ruler put next to it, at least up to the 10-inch mark. In fact, my argument is bolstered by the fact that the points end up being different. The point was that you can shift a whole ream of points down the line and it will never change the length of the line.

I have a feeling you think the mapping argument is crucial because, again, that’s how it works with finite sets. With finite sets, if you want to know whether two sets contain the same number of members, you enumerate them. You say: that’s 1, 2, 3, 4, 5 things in the first set. That’s 1, 2, 3, 4, 5 things in the second set. Yep, for every item I counted in the first set, there is an item that I counted in the second set. When I stopped counting the first set, I stopped counting the second–not an item before, not an item after. Enumeration and mapping only work because the sets are finite. They work because there is an end to the enumeration and the mapping that you will eventually attain, and when you do, you will know whether the one set has more members than the other, has less members, or is the same. If you find there are members left over in one set that can’t be mapped to members of the other set, that’s how you know the first set has more members. All this hinges on an end to the enumeration and the mapping. But if you enumerate an infinite set and you map each member to members of another set as you enumerate that set, you’ll never be able to say: well, it appears I’m done counting this set before the other. It appears there’s members left over in the other set that can’t be mapped. It doesn’t matter if you map only every second member, or every third, or every fourth, the result will be the same: it’ll go on forever, and skipping every second, third, fourth will never show that there’s fewer members in the latter set than in the former.

^ Now don’t say I avoid explaining why this leap from finite sets to infinite sets is a flaw.

Don’t you know about second infinity, Silhouette? Ask Magnus all about it.