>>13606566Not the same anon, but the incompleteness theorems do not entail that any set of axioms contain contradictions, what they any formal system that satisfies the Peano axioms must contain statements that are true, but not provable. Hence there are two basic problems with your claim:
(1) the completeness theorem only applies to formal systems that are strong enough to encapsulate Peano arithmetic. This is akin to grade school arithmetic. I.e. you get addition, subtraction, multiplication, exponentiation, variables, etc. However, there are many axiomatic frameworks weaker than Peano arithmetic that are in fact complete. E.g. basic propositional logic is complete. So is Presburger arithmetic, which is basically just the natural numbers with an addition operation.
(2) The incompleteness theorems do not generally make a claim about inconsistency (although their is a variant involving consistency). What they say is that there are true claims that can't be proven. Inconsistency, on the other hand, occurs when a system proves a statement as well as the negation of the statement, i.e. p and not p. It might help to think of incompleteness as "not proving enough theorems" and inconsistency as "proving too many theorems".
There is also another variant of the completeness theorem, the 2nd incompleteness theorem, which states that If Peano arithmetic is consistent, then it can't prove (about itself) that it is consistent. But again, this does not entail that Peano arithmetic is inconsistent.
pic unrelated
