AF. Counting All Rationals
Are there more fractions than counting numbers?
After answering last week’s puzzle we start to discuss Cantor’s incredibly profound question: Are there more fractions than counting numbers?
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strauss said,
April 9, 2009 at 11:27 am
A correspondent writes:
So everyone knows that there are an infinite number of numbers between any two numbers on the number line, but does this account for two numbers that are infinitely close to each other? While this seems like it should be the case, it also seems like it contradicts itself.
The trick is: What do you mean?
That is, words like “infinity” are not really precise notions out in the universe, but human attempts to get at specific themes and ideas. Philosophy, to a great extent, is the art of teasing apart all the different, subtle and delicate ideas that are tangled up in such a word.
Mathematics takes a very pragmatic approach; we define ideas, very precisely, aiming for a certain utility. The idea of “infinitely close numbers” may or may not make sense, depending on how we set up these definitions.
And paradoxes, in this light, aren’t so much puzzles about the real-world, but simply signposts, pointing to disconnects in our intuition, or problems in the definitions.
SO: in the usual definitions of the real number line, distinct numbers always have a positive distance between them; numbers that are infinitely close are in fact the same number.
In other number systems, such as Conway’s surreal numbers, numbers can have positive, but infinitely small distances between them, and there are still infinitely many numbers in this gap. But the magic all lies in the precise definitions.
Hope this helps,
Chaim