Think about R the set of real numbers. Are there any exceptions for any of its subsets being uncountable? If they exist, what are they?

Question

Professor says this question is a lot simpler than I think 

Answer 1

There are a lot of countable subsets:

 

Natural numbers: Trivial.

1,2,3,4,5,6,7,...

 

Integers: Sort by absolute value.

0,1,-1,2,-2,3,-3,...

 

Positive Rationals Rationals: Sort by sum of numerator and denominator.

 

0,1,2,1/2,3,1/3,4,3/2,2/3,1/4,...

 

Rationals: Like above, but after each positive number, put it's negative.

 

Algebraic Numbers: First list all irreducible polynomials with integer coefficients and then replace each one by all it's roots.

 

Computable Numbers: This one is difficult to give a specific method, but since every one is by definition possible to compute to arbitrary accuracy by a finite computer program, all you have to do is show that there are only a countable number of programs.

 

Note that Rationals, Algebraics, and Computables are also dense (between any two elements is another element).