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. "