Is 1 = 0.999... ? Really?

[Magnus Anderson]

I think you have to stop using the word "size". James spoke of 'degrees of infinity".

Emphasis is mine.

viewtopic.php?p=2614566#p2614566

An infinitesimally small element has no size, it’s not defined.

It has size, just not a specific size other than merely too small to measure.
infinitesimal = 1/infinite

viewtopic.php?p=2614807#p2614807

Car[di]nality is entirely about the quantity of elements in the set. It is false that the natural numbers and the even numbers have the same quantity of elements (albeit a common misunderstanding). I can prove that they are necessarily not the same size, but let's see your proof first (since you brought it up).

viewtopic.php?p=2615136#p2615136

Different size infinite sets can have the same summation limit. The limit DOES NOT specify the final summation.

viewtopic.php?p=2615491#p2615491

We have previously agreed that infinites can be of different sizes. One particular size can be ("may validly be ") set as a standard with which to compare others.

[..]

I am giving one particular infinite size a name, "infA", to be the size of the set of all integers. InfA does not have to be called a "number" or a "real number" or a "quantity" or anything other than a standard size (aka "infA elements") of the infinite set of integers.

[Magnus Anderson]

Obviously! That would take an eternity.

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.

[Ecmandu]

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.

[Magnus Anderson]

Isn't that mapping kind of arbitrary?

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.

You just took two identical sets and said: you know what... only the odd numbers from A map to B.

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.

Putting those questions aside, I can see how this carries over to my example of the two lines. Before removing the points, the points in each line are like the numbers in each set A and B. Then when I remove the odd points from the second line (call it line B) and move the remaining points into the gaps, that's like mapping every odd number in A to every number in B. Have I got that right?

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.

I think I'd prefer your statement that one cannot determine the size of infinite sets... admittedly, whether or not infinite sets have a size isn't clear to me

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.

So let's say we label the points in each line. Let's label the points in line A a1, a2, a3, etc. The points in line B are b1, b2, b3, etc. Now, with labels attached to them, the lines would look different after removing every odd point and moving the remaining points to fill the gaps (they weren't quite the same to begin with, what with line A having 'a' in the labels and line B have 'b', but let's ignore that for now). You'd see a1 paired up with b2, a2 paired up with b4, a3 paired up with b6, and so on. I suppose your point is that even without the labels, it's which point is paired up with which point that makes the difference. The first point in line A ends up paired with the second point in line B, the second point in line A paired up with the fourth point in line B, etc.

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.

It's as if each point has a special identity. If it's the identity of each point that matters, then the lines were never identical in the first place. Point a1 is not point b1 (even without the labels).

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

But none of this addresses what it means for line B to be "shorter" than line A after removing every odd point and moving the remaining points to fill the gaps.

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.

By the conventional definition, both lines would still appear to be just as long (or if you'd like, their lengths remain just as undefined, no reason to say one is shorter than the other).

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

I'm still looking for this unconventional way of defining "shorter". Did you want to grab the concept I proposed earlier? That "shorter" means there is nothing but gap at the "end" of line B whereas there is "more line" at the end of line A? Or is it the history of the lines? What they went through? In that case, "shorter" means line B, in its past, had half its points removed. <-- That could work to say line B is now shorter than its previous length, but it still raises some questions. Why did we decide to say the two lines were the "same" length in the beginning? And how could we tell that line B is now shorter than it used to be if we didn't see every odd point being removed? Much remains to be fleshed out with this definition.

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".)

[Ecmandu]

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

[Magnus Anderson]

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.

[Ecmandu]

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.

[Magnus Anderson]

Let me show you how hypocritical this example still is:

I suppose you mean contradictory.

Your objection to \(\frac{9.\dot9}{10}=0.\dot9\) is that at some "end" to the infinite recursion of 9s there's a spare 9 for \(0.\dot9\) that doesn't match to \(9.\dot9\)

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.

Yet somehow for your visual display of \(0.\dot9\)s, starting one "1 decimal place" after the other makes it shorter. Here the ends are equal, but previously they weren't.

What "ends" are equal? And what "ends" were previously not equal?

Convenient how you can judge sizes arbitrarily to fit your point, no?

That's precisely what you're doing.

You can arbitrarily match one-to-one correspondence however you like so that A can be any "size" larger or smaller than B and vice versa.

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.

For the bijection of \(\frac{9.\dot9}{10}\) and \(0.\dot9\), you insist on arbitrarily corresponding different numerical positions just to fit your point when you could do it another way because you naively think dividing by 10 literally does nothing more than shifting digits.

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

You can't define the infinite because infinite means boundless and define means to give bounds

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.

I've said this so many times and you've not once addressed the obviousness of this explanation, and you still insist I'm just "telling" you what to think as if the explanatory logic behind this obvious fact was merely some kind of subjective demand.

Yes, you are still telling me what to think.

And you actually think I'm being silly about the extensive proof that 1+1=2

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.

[obsrvr524]

I think you have to stop using the word "size". James spoke of 'degrees of infinity".

.
.

It has size, just not a specific size other than merely too small to measure.
infinitesimal = 1/infinite

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.

1) 0.333... is not a number - is a type and degree of infinity

But "standard math" holds to the word and concept of "infinite" to mean merely "endless", without concern for what degree of endlessness is involved. That distinction didn't gain authority in mathematics until around 1947 with Hewitt and still isn't being taught much to many even today. And because of that, many simple minded paradoxes are accepted as nonintuitive truths which are in fact merely poor semantic conflations (e.g. "They are both infinite therefore they are equal length").

It is always and forever merely up to what standard of minimal measure one chooses - what degree of infinitesimal is going to represent "1". If none is chosen, then a physical ontology cannot be formed because every infinitesimal distance could be infinitely divided such as to have no means to sum up any distance at all.

Ontology is a Choice. One must choose how many infinitesimals are going to exist between 0 and 1.

The distance can be divided infinitely, but as long as the infinitesimal segments are multiplied by the same degree of infinity (yes, infinity comes in degrees), the original distance is reestablished. And that resolves the issue of Zeno's paradox.

With a possibility being that degree of infinitely small

Absolute infinity cannot exist simply because no matter how great a measure is, more can be added. No matter to what degree an infinite measure is, a greater degree can be formulated.

[obsrvr524]

And you actually think I'm being silly about the extensive proof that 1+1=2

Reference?

[Ecmandu]

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.

[Magnus Anderson]

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.

I see.

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

[iambiguous]

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.

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.

[gib]

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.

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.

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.

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?

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.

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.

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.

And you don't see the problem with this?

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.

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?

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.

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.

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.

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.

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?

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

How would they appear to be just as long?

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.

And what does it mean that their lengths are undefined?

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.

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".)

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.

[Magnus Anderson]

And how do you determine the mapping between two sets?

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

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?

Due to logical consistency.

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.

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.

The first point in line A must be labeled something different form the first point in line B

That's unnecessary.

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.

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

And you don't see the problem with this?

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

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.

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.

The labeling doesn't matter because it's arbitrary

It does matter.

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

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.

They're infinite!

Infinite means endless. It does not mean "equal in size".

They remain just as infinite no matter how many points you remove!

They remain endless but their sizes change.

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".

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.

^ 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.

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

[Silhouette]

Let me show you how hypocritical this example still is:

I suppose you mean contradictory.

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.

Your objection to \(\frac{9.\dot9}{10}=0.\dot9\) is that at some "end" to the infinite recursion of 9s there's a spare 9 for \(0.\dot9\) that doesn't match to \(9.\dot9\)

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.

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.

Yet somehow for your visual display of \(0.\dot9\)s, starting one "1 decimal place" after the other makes it shorter. Here the ends are equal, but previously they weren't.

What "ends" are equal? And what "ends" were previously not equal?

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.

You can arbitrarily match one-to-one correspondence however you like so that A can be any "size" larger or smaller than B and vice versa.

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.

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.

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

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.

Convenient how you can judge sizes arbitrarily to fit your point, no?

That's precisely what you're doing.

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.

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.

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!!!

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.

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.

Yes, you are still telling me what to think.

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.

And you actually think I'm being silly about the extensive proof that 1+1=2

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'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.

And you actually think I'm being silly about the extensive proof that 1+1=2

Reference?

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.

[Magnus Anderson]

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.

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.

But even if the purple could be justified as starting after the green

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.)

You literally created identical sets, i.e. bijective by definition, \(A = \{1, 2, 3, \dotso\}\) and \(B = \{1, 2, 3, \dotso\}\)

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

All words obviously have definitions, but the things they refer to can defy definition - making the act definition questionable in the first place.

That's nonsense.

The signified has infinitude, but the signifier implies at least some finitude for it to be a word at all.

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.

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.

There is only one way that sets can be infinite.

[Magnus Anderson]

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?

\(0.\dot01\) 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.

[Magnus Anderson]

I criticised the use of "infinite" in reference to sets when any aspect of their construction is a product of finitude.

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

I mentioned how even the natural numbers have a finite starting point on the number line

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

never mind the line being finitely bounded in all other dimensions as well

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

being infinite in only one dimension in one direction

How can sets be infinite in more than one dimension and in more than 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.

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

In other words, any size of "infinite sets" is determined by their relative lack of finite constraints and not any "different size of infinity".

What does it mean that sets have "finite constraints"?

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".

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.

[Silhouette]

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 (\(10 \times 0.9 = 9\)).

Omfg I know why you put it there!! 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.

You literally created identical sets, i.e. bijective by definition, \(A = \{1, 2, 3, \dotso\}\) and \(B = \{1, 2, 3, \dotso\}\)

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

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.

All words obviously have definitions, but the things they refer to can defy definition - making the act definition questionable in the first place.

That's nonsense.

No it really very super isn't!

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"?

The signified has infinitude, but the signifier implies at least some finitude for it to be a word at all.

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, it's 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.

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.

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.

There is only one way that sets can be infinite.

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.

\(\prod_{n=1}^{\infty} \frac{1}{10} = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \dotso\) The infinite product has no last term. The end is merely in the symbol representing the infinite product.

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.

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.

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.

[obsrvr524]

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).

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.

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.

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.

[Magnus Anderson]

\(\frac{9 + 0.\dot9}{10}=0.\dot9\)
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.

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 \(0.\dot9\).

[gib]

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

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?

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.

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.

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.

The first point in line A must be labeled something different form the first point in line B

That's unnecessary.

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

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.

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.

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.

The labeling doesn't matter because it's arbitrary

It does matter.

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 the logic of finite sets carries over to infinite sets.

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

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.

They're infinite!

Infinite means endless. It does not mean "equal in size".

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).

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.

a level of stupid i have not seen in a long time.jpg (29.3 KiB) Viewed 389 times

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.

^ 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.

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

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.

[gib]

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

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

[MagsJ]

Still following..

I will adject when I see fit, but right now my money is still on Magnus, as man’s held his ground and not swayed from his thinking. No pressure though bro, ok? lol