Does "provable" include "proved by reduction to absrdity"?

Question

The incompleteness says that formal logic system (under certain condition) contains non provable TRUE sentence. It seems that "prove" means here "derive". Only TRUE sentence could be proved. If a sentence is FALSE, it cannot be proved, but the sentence of its inverse could be proved.

The reduction to absurdity can prove both TRUE and FALSE sentence. So does this mean that "provable" here doesn't include "proved by reduction to absurdity" ? Or in another words, the reduction to absurdity is not in the rules of inference of logic system ?

If "provable" doesn't include proved by the reduction to absurdity, could it be understood that "a non provable TRUE sentence" means that a sentence that is not provable but can be proved by reduction to absurdity that it's TRUE ?

Answer 1

If you prove a false sentence, whether with reductio ad absurdum or any other rules, you will have an inconsistent system, and everything will be true in such a system.

In classical logic, and many other logics, reductio ad absurdum is a valid proof rule, and, assuming that your system is consistent, it can only be used to prove true sentences.

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Reductio ad absurdum (RAA), in a formal system, is a rule of deduction of the following form. $$\begin{array}{c} \neg S \\ \vdots \\ \bot \\ \hline S\end{array}$$ The rule says that, if you from the negation of the sentence $S$ can derive a falsehood (which we denote by $\bot$), i.e. something that is always false (and which we informally call a contradiction), then you can conclude that your assumption that $\neg S$ is true must be false, so $S$ must be true.

Answer 2

The provable includes "proved by reduction to absurdity RAA".

It should be distinguished a logic system and a theory system. A theory system is based on a logic system by adding non logic axiom.

When a formal system contains enough rule of derivations, it can derive all TRUE formula (formula that is semantically TRUE). This is the completeness theorem.

What is surprising for the incompleteness theorem is that it shows that from a complete logic system, adding non logic axiom to make theory system, the completeness of the theory system changes when the theory system contains Peano arithmetic. The theory system will become incomplete.

So, the incompleteness is caused by the non logic axioms added, not due to the derivation rules it contains.