Don't want to pile on too much, but my attempt at an explanation:
> Simply flip the digits over the decimal point, so 0.14159 become integer 95141
They key here is that both 0.14159 and 95141 contain a _finite_ number of symbols/digits. There are infinitely many numbers like this, but each of those numbers are themselves finite.
The decimal expansion of pi, OTOH, is _itself_ infinite. The decimal expansion of any irrational number is infinite. There are infinitely many of these things, each of which are themselves infinite.
This is the key thing that causes uncountability. Each of the numbers has infinitely many "parameters" that can be tweaked. If you try to produce an infinite (countable) list that contains all of these infinite numbers, I can always tweak a different parameter for each number in your list and be guaranteed to have produced a number that your list missed.
> Simply flip the digits over the decimal point, so 0.14159 become integer 95141
They key here is that both 0.14159 and 95141 contain a _finite_ number of symbols/digits. There are infinitely many numbers like this, but each of those numbers are themselves finite.
The decimal expansion of pi, OTOH, is _itself_ infinite. The decimal expansion of any irrational number is infinite. There are infinitely many of these things, each of which are themselves infinite.
This is the key thing that causes uncountability. Each of the numbers has infinitely many "parameters" that can be tweaked. If you try to produce an infinite (countable) list that contains all of these infinite numbers, I can always tweak a different parameter for each number in your list and be guaranteed to have produced a number that your list missed.