I'm reading this book: https://www.amazon.com/Introduction-Logic-Methodology-Deductive-Mathematics/dp/048628462X/ref=sr_1_2?keywords=tarski+alfred&qid=1581605512&sr=8-2 and it states:
"A theory is called complete, if of any two contradictory sentences which are formulated by employing exclusively the terms of the theory under consideration (and of the theories preceding it), at least one sentence can be proved within this theory."
and then it says: "a discipline might be considered ideal, if it contains among its asserted statements all true sentences which are from the domain in question, and not a single false one. A deductive theory certainly falls short of our ideal unless it is both consistent and complete."
my question is, how can a theory have all true sentences and be complete? I don't know if I don't understand something in the definition, but saying "if of any two contradictory sentences ..., at least one can be proved" to me it seems that in this theory one of these sentences is false (since they are contradictory). Is it because it's an implication "if.. then.." and if the antecedent "of any two contradictory sentences" is false the implication is true? or what am I missing?
The two contradicting sentences don't have to be part of the complete theory, they only have to be "formulated by employing exclusively the terms of the theory under consideration (and of the theories preceding it)".
Meaning:
Given a theory $K$, such that $t_1,t_2,...,t_n$ are all the terms of $K$ and of $K$'s preceding theories, and for any given $\varphi$ such that $\varphi$ is a sentence that is formulated from $t_1,t_2,...,t_n$, $K$ is said to be "complete" iff it proves $\varphi$ or $\neg \varphi$.
Neither $\varphi$ nor $\neg \varphi$ need to be part of $K$, the requirement is for them ($\varphi$ and $\neg \varphi$) to be formulated from $t_1,t_2,...,t_n$.