This is a very vague question, but a question nonetheless. I am becoming increasingly more interested in what can be vaguely categorized as Mathematical Philosophy, or more specifically perhaps, Metamathematics, that is, the study of Mathematics itself! I.e. Theorems about theorems.
Examples of such theorems would be the Incompleteness theorems, Hilbert's tenth problem, the Continuum Hypothesis etc... I don't know what 'branch' of Mathematics to bracket these theorems into but I am sure that those reading this post know exactly what I mean.
My question is, can anybody recommend any books that would introduce me to this 'branch' of Mathematics? I don't mean Mathematical Philosophy for laymen, but a proper introduction to the theory.
Many thanks, Elie.