# Understanding the syntactical completeness

A formal system is syntactically complete if for each sentence (closed formula) $$\varphi$$ either $$\varphi$$ or $$\lnot \varphi$$ is provable.

A formal system is semantically complete if every sentence (closed formula) $$\varphi$$ which is tautology is provable.

What I wonder is

1. The Godel's first incompleteness theorem states that there is a sentence $$\varphi$$ in Peano's arithmetic which is true, but which is not provable and which negation is not provable, so it is at the same time syntactic and semantic incompleteness theorem, right? Is it right to say that the semantic incompleteness is more essential result since it implies the syntactic incompleteness under the assumption of soundness?

2. Consider the formula $$P$$ in propositional calculus where $$P$$ is a propositional variable from the signature. It is syntactically incomplete since neither $$P$$ nor $$\lnot P$$ is not a tautology, and by the semantical completeness of propositional calculus, we can conclude that neither $$P$$ nor $$\lnot P$$ are deducible. However is there a way to prove that neither of them are deducible syntactically, without the semantic argument?

3. Are there any examples of useful syntactical complete formal systems?