How can Godel's theorem apply to every formal system?

Question

How can Godel's theorem apply to any formal system of logic, if the truth of the theorem itself is only relative to the axioms and rule of inference that were used to generate it. In other words couldn't we just as easily select an axiomatic system in which Godel's theorem isn't true, and go about mathematics from there?

Answer 1

In addition to what has already been said, Gödel's theorem does not assert that any theorem in a given axiomatic system can be chosen to be either true or false, but that there are statements in that system whose correctness is indecidable in that system. Usually such statements are hard to find, and Gödel went through considerable effort to find one (later generations have formulated simpler procedures). Mathematically interesting theorems usually do not belong to this class of undecidable statements.
Furthermore, a statement being indecidable does not mean that it is "neither right nor wrong" in the common sense of the words, but rather that the axiomatic system in which it is formulated is not sufficiently powerful to decide that question. Accepting more powerful methods of proof (e.g. those not limited to a finite number of logical steps) can also change this situation.
Finally, Gödel's theorem is usually stated for axiomatic systems which are "complex enough to embrace arithemtic" (e.g. which contain the Peano axioms), because there are exceptional examples of very simple axiomatic systems which do not have Gödel's incompleteness property.

Answer 2

In theory: yes; in practice: not really. Gödel's theorem does not use any doubtful assumptions. The theorems can be carried out in simple theories of arithmetic such as PA, and most of the notions use only primitive recursion. So if you chose a system where Gödel's arguments cannot be carried out you will have a very weak system insufficient to demonstrate the other theorems we need for ordinary math. It is like eliminating prime numbers.