As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ.
Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities.
* How many natural numbers are there? Countable
* How many rational numbers are there? Countable
* How many numbers are solution to a polynomial? Countable
* How many numbers are the output of any turing machine (including programs that run forever, producing an infinite decimal)? Countable
* How many numbers are the answer to any maths problem anyone can write down (where the problem has at most a countable number of answers)? Countable.
If the set of all numbers any can express in any sensible way, and the solution to any problem any could ever have, is countable, why do we need the other uncountables?
Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities.
* How many natural numbers are there? Countable
* How many rational numbers are there? Countable
* How many numbers are solution to a polynomial? Countable
* How many numbers are the output of any turing machine (including programs that run forever, producing an infinite decimal)? Countable
* How many numbers are the answer to any maths problem anyone can write down (where the problem has at most a countable number of answers)? Countable.
If the set of all numbers any can express in any sensible way, and the solution to any problem any could ever have, is countable, why do we need the other uncountables?