Hacker News Reader

"consider the set of numbers that cannot be described using any finite collection of symbols" - isn't that set empty?

If "numbers" mean "real numbers" then I can describe easily "smallest positive number in the set" and "biggest negative number in the set" - so neither negative, 0 nor positive real numbers might be in the set.

View on HN

It's trivial to prove that the number you are describing doesn't exist:

Let x be the smallest positive real greater than 0. Since x > 0, x/10 < x, and x/10 > 0. But now we have a contradiction, since x was supposed to be the smallest positive real number. So, x doesn't exist.

Edit: note that this doesn't even require real numbers. Even for the rational numbers, with common definitions of less-than, there is no smallest positive rational number >0, per the same logic.

What is the "smallest positive number" in the set {x | x in reals and 0 < x < 1}?

Reply on news.ycombinator.com