I am taking a course where we are covering a bit of logic, and I am trying to understand a some nuances of Gödel's theorems of completeness and incompleteness.
Q1) Is it correct to say that Gödel's completeness theorem refers to the completeness of the deductive system where as Gödel's incompleteness theorem refers to, an unrelated, concept of completeness of a set of axioms?
I am curious because I have been told that Gödel's incompleteness theorem states that a deductive system is either unsound or incomplete. Which I believe to be wrong.
Q2) In first order logic it is my understanding that you cannot reference a sentence inside of itself. Such as "This sentence is not true" or "This sentence has no proof".
However, I am given the sentence "$\alpha(j, A) = \forall i$ $\ i$ is not the Gödel number of a proof of the sentence whose Gödel number is j, where the proof uses only premises in $A$". ($A$ is recursively enumerable set of axioms).
Then we let $\sigma = \alpha(\#\sigma, A)$ where $\#\sigma$ is the Gödel number of $\sigma$. Can we really say that $\sigma$ is not self referential in this situation? How is this different from the statement "This sentence has no proof"? This is from Introduction to Artificial Intelligence by Russel and Norvig.
Q3) Finally, is the sentence $\sigma$ independent of the set of axioms $A$, i.e. is both $A\wedge \sigma$ and $A \wedge \neg \sigma$ both satisfiable?
EDIT: The following argument is nonsense. Just leaving here to recall my train of (incorrect) thought.
It seems like it must be the case that both are satisfiable. Otherwise, if $\sigma$ is always a true statement, then that implies $A \models \sigma$, so $A \vdash \sigma$ by completeness, so now we have a proof, so $\sigma$ is false.
But if $\sigma$ is always false, then $A \wedge \neg \sigma$ is unsatisfiable, so $A \models \sigma$. Then $\sigma$ is true.
It has been a while since I have thought about this, can anyone clarify a bit?