- Mar 29, 2010
- Androgyne; they/them
- Basic Beliefs
- Natural Philosophy, Game Theoretic Ethicist
And this exercise is similar to setting aside the odd numbers for the evens, setting aside the 1's for the 1/2s, setting aside for the 2/3's and the 1/3's, etc, each also ending up the same setHaving more things between any two things isn't what makes a set uncountable. Between any two rational numbers there are more rationals; for instance (A + B) / 2 is in between A and B. But the rational numbers are a countable infinity.Countable vs uncountable infinity is generally seen as the difference between a series of numbers that can each be enumerated one to the next in series, and a series in which this is impossible because there are things between each of the things, and you can always find another thing to set between any two.
To get to uncountable you need something that doesn't order as a count but rather as an evolving operation on a count, as per numbers like pi, where 3, into 1, into 4....
So you were right insofar as I described it badly.
We don't end up uncountable until irrational numbers start happening, numbers which are defined in terms of a recursive or internally iterative process.