I have problems understanding the difference between a categorical theory and a complete theory. My intuition says that every valid complete theory must be categorical. Is it true?
Clarification: by "complete theory" I mean "maximal consistent theory". Refer to the Wikipedia article: http://en.wikipedia.org/wiki/Complete_theory.
Here's what I understand:
A theory is a tool to accept or reject structures.
A theory can be trivial, that means it accepts any structure. Its axiom set evaluates to logical truth.
A theory can be inconsistent, that means it rejects all structures. Its axiom set is contradictory and evaluates to logical false.
A theory can be categorical, that means it accepts exactly one structure.
A theory can be complete, that means it has something to say about any sentence, or incomplete, that means it leaves some sentences out.
Now, are the following claims correct?
A trivial theory is incomplete, in fact it does not say anything about any sentence.
An invalid theory is complete, in fact it says "yes" to every sentence (and to its negation too).
A categorical theory is complete. That means, categoricity implies completeness.
Now, can there exist a theory that is complete and valid but not categorical? If yes, then how? If a theory is to have two different models, then it must leave some sentences undecided, so the models can differ in that point. Is it right?