Wednesday, September 3, 2014

Forming an infinity step-by-step

Suppose I am assigned to write a book with infinitely many pages. Luckily for me, the past is infinite. Then I could have accomplished the task on a leisurely schedule by having written a page every decade.

But the odd thing is that then the task will then have always been finished! For in every past, present or future decade, it is true that I've already written infinitely many pages.

Yet the task of writing a set size of book is surely a paradigm of a task done step-by-step. And it is a central feature of doing a task step-by-step that the task wasn't always completed. So the task of writing an infinite book wasn't always completed, and yet it was always completed, a contradiction.

Hence, the task of writing an infinite book a page at a time is impossible.

The best response I can think of is to deny that writing a book of a given size a page at a time is always a task done step-by-step. It's true for finite sizes, but need not be true for infinite ones. Maybe the problem is that it's not a possible task (this thought is inspired by Richard Gale's comments on Zeno paradoxes).

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