# Is Countable set always countable?

Why some sets are countable and some are not countable? Say regular set are countable but how (0+1)* will be countable? It is an infinite string, then how it could be a countable set? How the set of all non-decreasing functions from N to N are countable? How the set of all finite partitions of N are uncountable?

• I'm voting to close this question as off-topic because it is a question about pure mathematics that has no computational content. – David Richerby Sep 29 '18 at 13:05
• This lives on Mathematics. – David Richerby Sep 29 '18 at 13:05
• Regular sets need not be countable. The set of all non-decreasing functions from $\mathbb{N}$ to itself is also not countable. – Yuval Filmus Oct 3 '18 at 2:52
• @YuvalFilmus Recursive Enumerable set is countable. Right? and all other language like regular, CFL, CSL, Rcursive language can be uncountable. – Srestha1 Jul 10 at 4:53
• The language of all words is countable. – Yuval Filmus Jul 10 at 14:59