i think above answers are equallt nice ,here is my prespective, i will take an example
say alphabets are {a,b} then all the strings on the alphbaets can be given as ( a + b )*
now since the above langauage is countable ( as each strings can be enumerated by a natural number )
now the way of proceeding will be , we will show that there a countable TM , but uncountable language and if we are able to show the above result then we can deduce that , as TM recognises REL so REL are countable , and NON-REL is subset of a uncountable set which is uncountable thus all NON REL are uncountable.
=> power set of natural number is uncountable , and (a+b)* is countable so power set of a countable set is uncountable here power set of (a+b)* is set of all the language generated over {a,b} , which are uncountable. now TM are countable ( as TM machines have finite representation , hence they could be represented by countably finite set of configuration , also called encoding of TM as (0,1). } ,