Comment by kerwioru9238492
3 days ago
My understanding is that for any system of axioms strong enough to encode arithmetic, you can have at most two of these three properties:
1. Complete (for any well formed statement, the axioms can be used to prove either it or its negation)
2. Consistent (can't arrive at contradictory statements ~ arriving at a both a statement and its negation )
3. The set of axioms is enumerable ~ you can write a program that lists them in a defined order (since the workaround for completeness can be just adding an axiom for the cases that are unproven in your original set)
If my understanding is correct, I believe your explanation is missing the third required property.
It's also important to point out that if we cant prove a statement or its negation (one of which must be true) then we know there are true statements that are unprovable. This is a much stronger of a finding than "Godel's first incompleteness theorem says that in any axiomatic system (sufficiently complex) there are theorems that are neither always true nor always false. "
It's also important to point out that if we cant prove a statement or its negation (one of which must be true) [...]
Is that true, could it not be neither, i.e. independent of the axioms? Or is this assuming completeness which rules out independent statements?
I just want to be a bit pedantic here (but this is logic after all...), and point out that in point 1 above you are talking about syntactical completeness, and not semantical completeness, which is the kind of completes Gödel proves in his first completeness proof. I think people are often confused because of this overloading of the word. And it is about sentences(a formula withouth free variables), not any well formed formula.