>>10468211I never liked this explanation. To someone uninitiated, it seems like it presupposes that there is an objective truth value to every statement. It's confusing, because it doesn't seem obvious a priori whether something like the Continuum Hypothesis should be true or not.
IMO, a better way to explain it:
"For certain sets of axioms (recursively enumerable, first order systems capable of modelling the natural numbers) there exists a statement that is true in some models and false in some (if the system is consistent)".
Just to give some intuition, here's a dumb example of what it means for a statement to be "true in some models and false in some". Note that it's not actually connected to Gödel.
Given the axioms of a ring(
https://en.wikipedia.org/wiki/Ring_(mathematics)#Definition), does there exist an element x such that x^2=-1? The answer is clearly, it depends. In the complex numbers it's true, but it's false in the reals, for example.
So we have at least one model where the statement is true, and at least one where the statement is false. So clearly, the statement is not provable from the axioms of a ring.
What Gödel's theorem says is that for "sufficiently powerful axioms" there are always, inevitable, statements like that (assuming consistency).
An inconsistent system, on the other hand, is a system capable of both proving and disproving (simultaneously) every statement you can possibly form. So it's completely useless.
I also want to mention Gödel's completeness theorem, another theorem of Gödel which is just as important but not mentioned as much. What Gödel's completeness theorem states is that the statements provable from a set of axioms are PRECISELY the statements that are true in EVERY model.
