TITLE: Why do people say Gödel's sentence is true when it is true in some models but false in others?
QUESTION [24 upvotes]: I am a beginner, so this question may be naive.
Suppose we have a (sufficiently strong) consistent first order logic system. Gödel's first incompleteness theorem says there exists a Gödel sentence $g$ which is unprovable, and its negation is also unprovable. By Gödel's completeness theorem, $g$ can't be a logical consequence of the axioms, which means there are models of the system that makes $g$ false. So my question is:
then why do people say $g$ is true when viewed outside the system?
PS: apparently there are "non-standard models" that makes $g$ false according to wikipedia, then why don't people say $g$ is true in standard models, which is more accurate? Also, do non-standard models work with natural numbers anymore?

REPLY [5 votes]: In your question, you start off saying "Suppose we have a (sufficiently strong) consistent first order logic system. Gödel's first incompleteness theorem says there exists a Gödel sentence  which is unprovable...".
You have made here a supposition, that the logic system you are studying is consistent. This supposition is the very one that the logic system you are studying may not be able to prove. You, having made that supposition, are therefore in a position to claim certain things to be true which the logic system itself may not prove.
This is how it can come to be that you claim something is true even though the logic system you are studying may not prove it. (And to say there are models of a logic system where a sentence is false is just a roundabout way of saying that the logic system does not prove the sentence.)

Keep in mind, Gödel's incompleteness theorem can be framed without making this consistency supposition in the first place. Indeed, it would be much easier to see the matter clearly and avoid such "Moore's paradox" issues if people ordinarily framed the Gödelian phenomenon without that supposition. "IF this logic system is consistent, THEN this logic system does not prove its Gödel sentence". "IF this logic system is consistent, THEN this logic system does not prove its own consistency". This is the Gödelian phenomenon, and this sentence is readily provable (aka, true in all models). There is no danger of paradox once phrasing things in these terms, even if you make the natural choice to consider the same logic system as both metatheory and object theory.
The best attitude to take about the Gödel sentence by default is to refrain from calling it true and to refrain from calling it unprovable (in the sense of unprovability in the particular logical system it references). Instead, simply note that its truth entails its unprovability and vice versa, whether or not it is true and whether or not it is provable. In some particular contexts, you may go beyond that, but in general, that entailment is all there is to say.