So to assume that it is infinitely divisible is to assume that something is capable of dividing it infinitely. — Metaphysician Undercover
I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous. — aletheist
What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical, and that any attempt to save continuous motion from Zeno's paradox by referring to being able to calculate the sum of a geometric series misses the point. — Michael
This is the assumption that I'm showing to be false. Each movement from one point to the next is a tick. — Michael
If motion is discrete, it's not motion as we understand it to be. As object A "moves" from discrete point 1 to point 10, what is the time lapse between 1 and 2? Does A go out of existence during the lapse, and how do we claim A maintains identity during teleport and reappearance?
You can't just offer discrete movement as a solution to the paradoxes associated with analog movement without also explaining how discrete movement really works. It might be there's no coherent explanation to something as basic as movement, just like there's not with causation.
Anyway, discrete movement is an obvious adoption of the computer graphics model imposed on reality. Identity of a computer graphic over time is preserved by the underlying programming, which is a quite literal deus ex machina. If we're going to insert Deus, I suppose anything is possible, including analog movement. — Hanover
But I'm not really interested in defending the notion of discrete motion. What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical, and that any attempt to save continuous motion from Zeno's paradox by referring to being able to calculate the sum of a geometric series misses the point. — Michael
Indeed. I do not see the relevance of being 5'11", unless as part of an argument to say that Max Planck was infinitely tall or something, or couldn't grow, and was born fully formed. — bert1
I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. — aletheist
That's right. It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. Why would you accept the contradictory premise, that something which is impossible to count is countable? That makes no sense to me. This is the basic nature of infinity, that it is not countable. To believe otherwise is very clearly to believe a contradiction. The notion of infinity may be useful, but it's a fiction, a useful fiction. — Metaphysician Undercover
Countable infinities are precisely those which can be put into one-to-one correspondence with the integers. This is a definition, and no, no one expects you to count them all. — tom
That's the point, they are not countable, so to call them "countable" is just a name, a label, it doesn't mean that they are actually countable. You might differentiate natural numbers from real numbers by saying that one is countable and the other not, but that's just a name, in actuality neither are countable. — Metaphysician Undercover
Try counting the real numbers between 0 and 1. — tom
It can't be done, but that doesn't mean that the natural numbers are countable. Neither real nor natural numbers are actually countable, because of the nature of infinity. One has no beginning point, the other has no ending point, but neither, as an infinity, is actually countable. — Metaphysician Undercover
Do you think you might be able to count a subset of the integers? — tom
Yes of course, but a subset of integers is not infinite. The difference here is with respect to the thing being counted, what is within the set, real numbers versus integers, one is assumed to be divisible, the other is not. It is not a difference in the infinity itself. With respect to the infinity itself, one is no different from the other. — Metaphysician Undercover
OK, so you can count integers, but you cannot count the real numbers, even in a tiny subset. There is an uncountable infinity of reals within any subset - hence it is a continuum. The countable infinities do not have this property. They are different and one is at least infinitely bigger than the other. — tom
No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity? — Metaphysician Undercover
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