Set theory cardinality question

[klabatiba]

Can anyone please give a really explicit proof (omitting no steps) and with as simple words as possible that any infinite set can be writtern as the union of disjoint countable sets?

Thank you.

 

[Preno]

How about something like this:

Every infinite set X has a countable subset (this is a theorem of ZFC). We'll construct a function f from an initial set of ordinals to disjoint countably infinite subsets of X such that X its range. Let f(0) be any countably infinite subset of X = X_0. Then define X_{i+1} = X_i \setminus f(i), X_i = \displaystyle{ \cap_{j<i} X_j} for a limit ordinal i and let f(i) is a countably infinite subset of X_i (the possibility of making these choices collectively relies on the Axiom of Choice). The sequence must eventually end with a null set - otherwise, the set X would be larger than all cardinals.

 

[Dragonfall]

Well, any set is a union of singletons containing exactly one element from the set. A union of all these singletons (which are finite thus countable) gives you that set.

 

[Elucidus]

I mention only that however the countable sets are constructed, the union must consist of an uncountable number of those sets.

--Elucidus

 

[Preno]

Well, obviously, countable sets cannot be expressed as an uncountable union of disjoint countable sets (assuming by countable you mean countably infinite, or at least non-empty).