## Saturday, April 21, 2012

### Completing the infinite

Anthony writes "An infinite procedure is, by definition, a procedure which can never be completed". Well no. By definition, an infinite procedure is one that can never be completed in a finite amount of time. To say it can never be completed at all begs the question, for it presumes that all processes are finite.
Cantor: If one considers the arguments which Aristotle presented against the real existence of the infinite (vid. his Metaphysics, Book XI, Chap. 10), it will be found that they refer back to an assumption, which involves a petitio principii, the assumption, namely, that there are only finite numbers, from which he concluded that to him only enumerations of finite sets were recognizable. [Punktmannigfaltigkeiten § 4 p. 104-5]
§ 183 Of all the philosophers who have inveighed against infinite number, I doubt whether there is one who has known the difference between finite and infinite numbers. The difference is simply this. Finite numbers obey the law of mathematical induction; infinite numbers do not… It is in this alone, and in its consequences, that finite and infinite numbers differ.

The principle may otherwise be stated thus: If every proposition which holds concerning 0, and also holds concerning the immediate successor of every number of which it holds, holds concerning the number n, then n is finite; if not, not. This is the precise sense of what may be popularly expressed by saying that every finite number can be reached from 0 by successive steps, or by successive additions of 1. This the principle which the philosopher must be held to lay down as obviously applicable to all numbers, though he will have to admit that the more precisely his principle is stated, the less obvious it becomes. [The Principles of Mathematics]

Anthony said...

"By definition, an infinite procedure is one that can never be completed in a finite amount of time."

Do you mean a finite number of steps?

William M. Connolley said...

Anthony alludes to a problem with your defn "Well no. By definition, an infinite procedure is one that can never be completed in a finite amount of time."

For example, consider the procedure "Move from 1 to 0 at a uniform speed of -1, crossing the points 1/n (n=1, 2... as you go)".

This procedure can be completed in a finite time, but contains an infinite number of steps. It is, of course, basically Zeno's paradox.

You're getting muddled. Infinity has nothing, intrinsically, to do with time.

Edward Ockham said...

Well, change the word 'time' to 'steps'.

Edward Ockham said...

I was assuming that each of the steps was finite, of course, so it was correct anyway.

Anthony said...

>> I was assuming that each of the steps was finite, of course, so it was correct anyway.

Then don't change it. Instead, explain how a process can be completed, but cannot be completed in a finite amount of time. Or give an example or something.

Is a sisyphean task one that can never be completed, or one that can never be completed in a finite amount of time?

Anthony said...

Anyway, I suppose I was also assuming that each of the steps would take a finite amount of time to complete.

So, an infinite procedure is a procedure with an infinite number of steps. And if the steps themselves are defined properly, such as in Zeno's paradox, then the procedure can be completed.

Hmm...

William M. Connolley said...

That looks wrong to me. It certainly isn't the defn I was taught, though I'm prepared to believe that there are many equivalent defns. "Can be put into 1-1 correspondence with a proper subset" is the defn of infinite I was taught (and anything not infinite is finite).

Its a better defn because more general; you don't need to have defined number, or successor.

But according to wik your defn is indeed the usual one; mine is http://en.wikipedia.org/wiki/Dedekind_infinite it seems. And they aren't quite equivalent.

Edward Ockham said...

>> I suppose I was also assuming that each of the steps would take a finite amount of time to complete.

Ah well I was assuming each finite step took exactly the same time.

Anthony said...
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Anthony said...

>> Ah well I was assuming each finite step took exactly the same time.

So you think a process can be completed, but can't be completed in a finite amount of time?

That makes no sense to me.

"Infinite number of steps", if the steps grow infinitesimally small, okay. I guess it's okay because the steps are imaginary. They are just divisions which imagine to have taken place.

But I don't see how a procedure can be completed, but can't be completed in a finite amount of time. Completed by whom?

Edward Ockham said...

I posted about this last year. I'm off for a bit but will try and find the post later. Cheers.