Saturday, September 26, 2009

Are most numbers big or small?

Consider the following statement:
For any fixed positive integer threshold K, there exists a finite number of positive integers less than K and an infinite number of positive integers greater than K.
One can plausibly interpret the above statement in two opposing ways.

1) Most numbers are big. Given any threshold K, there are always many more numbers bigger than K as opposed to less than K. So if we define big using some kind of absolute threshold, then most numbers are big.

2) Most numbers are small. For any number K, there are always more numbers bigger than K as opposed to less than K. So if we define big using some kind of percentile threshold, then most numbers are small.

Just another little quirk of them good old countably infinite sets.

2 comments:

mark said...

I always found the strangest thing about infinite sets is the property (which Dedekind took to be their defining property) that an infinite set can be put into a 1-1 correspondence with a proper subset of itself. For example, there are as many odd/prime/square/powers-of-ten natural numbers as there are natural numbers.

It's almost like a flashing neon above zero warning us to abandon all intuition ye who enter here.

Yisong said...

Infinity can be counter-intuitive like that =)