Mathematical logic: definition of a (maximal) theory in intuitive terms Could someone explain what a theory is in the most intuitive way possible? Also, what does it mean for a theory to be maximal? Is this the same as being complete? 
Thank you very much in advance for your help.
 A: In most formal-logical contexts, a theory is simply any set of formulas (sometimes a set of sentences), which you intend to view as axioms -- that is, you're going to be concerned either with either what you can derive from them, or about its models, or possibly both. That intention is the only thing that distinguishes a "theory" from a random set of formulas.
Some authors say that a theory is a logical vocabulary together with such a set of formulas; this is sometimes more intuitive in addition to technically convenient (especially for speaking about models and categoricity), but much of logic can be developed without needing that kind of pedantry, hence the difference between authors.
A "maximal" theory is not quite a standard technical term, but the most sensible technical meaning would be a consistent set of formulas (in a particular language) which has no consistent proper superset. This is a stronger property than being complete; a complete theory merely has to imply one of $\phi$ and $\neg\phi$ for every closed $\phi$, but a maximal one actually has to contain one of them as an axiom.
