Countable and discrete are separate concepts. The Cantor set, for example, is <A HREF="http://math.stackexchange.com/questions/210735/is-the-cantor-set-made-of-interval-endpoints">discrete, but not countable</A>.

As for discretizing space-time, the only theory I'm aware of that takes this approach is known as <A HREF="http://www.einstein-online.info/elementary/quantum/loops">loop quantum gravity</A>. <A HREF="http://www.superstringtheory.com/blackh/blackh4.html">String theory</A>, if my passing understanding is correct, does not discretize space-time in this way.