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.