Well, I’ve been doing a bit more than just asserting. I gave some counter-examples as a means of proof by contradiction:
I certainly would not have accepted that train B is longer because its carts are longer. Imagine two infinitely long sticks. They’re both infinitely long. For all intents and purposes, equal. Now imagine chopping up one stick into 90 inch long segments and the other into 93.75 inch long segments (what an odd number to choose). Is the one with 93.75 inch long segments suddenly longer?
To say each resultant queue is half the size of the original queue is to say it’s only half as long, or “shorter”. I just don’t know how to make sense out of that. For something to be “shorter” in length is to imply it has a beginning and an end. You’d have to imagine putting it next to something else and observing that it ends before the other thing ends. But we both agreed that the two resultant queues are still infinite. To me, that means I can’t imagine putting those queues next to the original one and seeing that they’re shorter. They would still seem to be the same length.
To say you’re two steps close to something that’s an infinite distance away is to imply the distance between you and that something has gotten shorter. But that implies there is an end to the distance, a point you are coming close to…
Not exactly proof that (\frac{\infty}{50}) = (\frac{\infty}{60}) = (\frac{\infty}{70}), but definitely proof that (\frac{\infty}{50}) > (\frac{\infty}{60}) > (\frac{\infty}{70}) can’t be true.
I also tried to explain why you have no right to switch between contexts when talking about infinity:
I don’t deny that there can be different orders of infinity (which is what I think James is getting at), but you don’t get there just by adding 1 to infinity. It’s more complicated than that. Take the example of the two parallel lines Ecmandu brought up. He says that since there is an infinite number of points in the first line, adding the second line, which also has an infinite number of points, doubles the number of points. Whether the arithmetic works like that or not (I don’t think it does), that’s not an example of a higher order of infinity. A higher order of infinity is more like a plane compared to a line–something you arrive at by compounding an infinity of infinities. The lines are only infinities of points, not infinities of infinities. But a plane is an infinity of infinities because it is an infinity of lines which in turn is an infinity of points.
The idea is like this: infinity, if you want to imagine it as something that you can somehow reach, is a “transcendent” object. To get there, you have to transcend all finite things (in the case of numbers, all numbers). It’s impossible, just like transcending space and time is impossible for physical beings like us, trapped within space and time. No matter how high you count, you’re no closer to infinity, just as no matter how far through space you travel, you’re no closer to being outside space. But if you want to suppose you somehow could reach infinity, you can imagine skipping the journey of counting (or traveling through space) and magically arriving there. In that case, you must objectify infinity–meaning you must now think of it as an object–i.e. a finite thing–this is your new unit, your new building block, your new fundamental particle in a higher universe–it is your new point. It’s like when you transcend all points in the line, you get the line itself. You can then treat the line as the new unit and start over adding lines together. Now counting consists of counting these lines, these infinities, and the new infinity to strive towards is the plane, the new transcendent object in this higher universe.
Going back to counting points, as in the case of counting up the points in the second line, is to go back below the first order infinity. You may think of it as going back to a different universe (i.e. a different line), but this is not the same as a different order of infinity. Adding the first infinity to the first point in the second infinity is not valid. It does not equal
∞ + 1. It’s adding apples and oranges. The infinity and the point are not only completely incommensurate objects, but they, in a sense, don’t even exist relative to each other (that’s why infinity is “transcendent”–it is “beyond” the universe of points–the point relative to the infinity could be thought of as the infinitesimal). You can add up objects in a box, but you cannot add the box to the objects (for example, 2 apples + 3 apples = 5 apples; but what about 2 apples + the box the apples came in? What does that equal?). But you can add several boxes together. The arithmetic works only in the same universe, not across universes.
^ But given your lack of response to this, I think it went over your head.
^ That, and I did argue earlier in this thread that infinity is not a number, and therefore arithmetic doesn’t necessarily apply to it. I know you can take the symbol (\infty) and plug into mathematical equations and do algebra with it, which is probably what you’re getting at when you say you can do arithmetic with it, but (\infty) in that case plays the role of a variable, an unknown, not an actual infinite quantity; you can put whatever you want in it’s place and the same rules would apply: (\alpha)/60, (\phi)/60, /60. (\infty) in this case doesn’t mean “the highest number possible”, it means “some undetermined quantity”… but it has to be a quantity, otherwise it doesn’t apply. The only sense I agreed with you that you can do arithmetic on (\infty) is if you treat (\infty) as a unit (where the quantity would be 1… like 1cm ← cm is the unit, 1 is the quantity). This is why I said you shouldn’t switched contexts willy-nilly–you’re switching from saying “1 of something symbolised by (\infty)” to “and infinity of something else (like train carts)”. When you do that, the math suddenly doesn’t work. (\infty) in this case is not simply defined as “an infinite number of things”, it’s defined as “some undetermined thing” (like a variable). But if you want to say you can do arithmetic on infinity (as an infinite number of things), I deny your right to do so.
And you’re also claiming that the word “shorter” implies an end. Again, without anything to back it up.
That’s pretty damn basic. I’d even say definitional… as in, I never thought you would have asked for proof given that that’s essentially how we define “shorter”. If Max is shorter than Harry, that means the tip of Max’s head is below that of Harry’s. If neither has a tip at the top of their head (because they’re infinitely tall or whatever), what does it mean to say one shorter than the other?