[alalv]

If you pick a number between 0 and 1 its going to either be a rational number, or is going to use a known irrational number (pi or sqrt(5)). In both cases it can be guessed in a finite number of guesses.

[alexmolas]

That's not true. The set of numbers between 0 and 1 is uncountable, which means that has a cardinality bigger than the natural numbers, ie: you can't build a bijection between [0, 1] and the natural numbers. Therefore, to fully specify any number in [0, 1] you'll need an infinite number of guesses.

[barrucadu]

There's an uncountable number of numbers between 0 and 1, but you, a human, can only pick from a countable subset of those.

[googlryas]

But the number picker specified the number in their head to themselves before the game began.