Wednesday, 14 December 2011

Book of Sand

Thoughts on the Book of Sand.

This occurred to me: open the book on a certain page, shut the page again, and you will never again be able to open at your earlier page. More: the pages cannot be numbered--there are an infinite number of them between the front and back boards; which means that, open it roughly in the middle, there are an infinite number of pages on either side.

But (and this is the part that troubles my head): let's say the book is an inch thick, and you took a half-inch paintbrush and painted the ends of the first half of the stack of pages gold. Afterwards, and whilst it would still be perfectly impossible for you to open the book at the same page twice, you could be 100% sure of opening the book in the first half. This puzzles me, because it suggest (counterintuitively) that it is possible to narrow down an infinite range of options. My sense is that this is no paradox: that half of infinity if still infinity, but it feels weird. It prompts me to ask: how far is it possible to narrow things down like this?


Mahendra Singh said...

An inversion upon Zeno's paradox of Achilles and the Tortoise … perhaps one could be embedded within the other for even more dizzying fun?

Gareth Rees said...

You've hit upon one of the foundational difficulties in set theory, which is solved by the concept of measure.

Sadly, the Wikipedia article is a bit high-level, so I'll sketch the background for you.

One of the breakthroughs in the analysis of real numbers is the idea that you can think of a numerical interval (like the interval between 0 to 1) as a set consisting of the numbers in that interval, instead of (as before) some kind of geometrical object like a line segment.

You can do lots of things with set-theoretic approach, but there's a stumbling block: you need to be able to reconstruct the notion of the length of an interval, which was straightforward in the geometric framework, but not so much in the set-theoretic framework, because an interval is an infinite set, and for infinite sets the notion of size is tricky, as witnessed by the paradoxes of set theory such as Galileo's paradox of square numbers, or Aristotle's paradox of the wheel.

"Measures" are ways of assigning "sizes" to sets of real numbers that act like lengths for intervals, but extend to much wider ranges of sets. There are lots of different kinds of measure: Lebesgue measure is one of the simplest (and historically one of the earliest).

Measure solves your Book of Sand conundrum: your gold-painted subset of the book may have an infinite number of pages, but it only has half the measure of the whole book.

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