Why do people say Gödel's sentence is true when it is true in some models but false in others? 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?
 A: 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.
A: When we say the Gödel sentence is true, we mean exactly the same thing as when we say the Fundamental  Theorem of Arithmetic is true, or Fermat’s Last Theorem, or any other theorem in mathematics.  We mean that we’ve proven it, using our standard consensus principles for reasoning about mathematical objects.  And when we talk about natural numbers — as in FTA, FLT, or the Gödel sentence — we mean the actual natural numbers, not arbitrary models of PA.
With FTA or FLT, we don’t usually even question that.  The reason we look at Gödel’s sentence in other models of PA isn’t because of any difference or subtlety in what the statement means — it’s just a difference in why we’re interested in it.  That difference comes back to the question of what principles we’re using in our proofs.
Most of the time, we just take those “standard principles” as an implicit background consensus, and don’t mention them explicitly.  But with our logicians’ hats on, we may want to be more explicit about them.  Most of the time, they’re assumed to be ZFC set theory or something closely equivalent.  So we can refine our statement that “the FTA is true” (or FTA, the Gödel sentence, etc) to “in ZFC, we have shown the FTA is true”, or more formally “ZFC proves FTA”.
And then to further refine it, we can ask: did we really need the whole power of ZFC, or does some weaker logical system suffice?  So we can ask whether these theorems are provable in PA, or any other logical theory T that has a way of talking about natural numbers.  And only then can we start asking about whether these statements may hold in some models of T and fail in others.  Which is an interesting question — and especially because in the case of the Gödel sentence, we can show it holds in some models and fails in others — but it’s very much a secondary one, and doesn’t affect the original primary meaning of the statement.  And it depends entirely on what theory T is under consideration.
The one subtlety to note here is that if we’re talking about “PA proves FTA” and “ZFC proves FTA”, these can’t quite be formally the same statement “FTA”, since one must be written in the formal language of arithmetic, the other in the language of set theory. What’s happening here is that the ZFC-version of “FTA” is an translation of the PA-version of FTA in ZFC, using ZFC’s set of natural numbers.  This translation is what “the standard model” means.  But it’s just part of giving a more refined analysis of the logical status of these statements — it doesn’t mean that every time we do any elementary number theory in ZFC, we should feel obliged to add “in the standard model”.  The whole point of a standard model is that it’s standard — it’s just giving the language of arithmetic its usual meaning within ZFC (or other ambient foundation).  You can equally well take “FTA” to be the PA-statement and view the ZFC-version as its interpretation under the standard mode, or take “FTA” to be the ZFC-statement and view the PA-version as a transcription of it into the language of arithmetic.  The former is more common in logic, but the latter is arguably closer to mathematical practice.
So overall: It’s completely accurate to just say “The Gödel sentence is true”, in the same sense that we mean when we say any other mathematical statement is true or false.  But if we want to refine that statement to a sharper one, then what we should say is “ZFC proves the Gödel statement [in the standard model of arithmetic].” — the part that really sharpens the statement is specifying “ZFC”, not the mention of the standard model.  Similarly, when we say that it is unprovable (or fails in some models), we need to be clear which theory we’re talking about provability in, or models of.
Edit.  I’ve assumed we’re talking of the Gödel sentence for PA, or some similar theory of arithmetic; but the same applies with the Gödel sentence for ZFC, or any other theory $T$. In Gödel’s theorem, we assume $T$ comes equipped with an interpretation of the language of arithmetic, and its Gödel sentence $G_T$ is a priori a sentence of arithmetic, that then gets (in the proof of Gödel’s theorem) interpreted into $T$. So again what it means when we say $G_T$ holds is no different in principle from what it means when we say FTA or FLT holds — it means “reasoning in the normal mathematical way, we can prove $G_T$ holds (in the natural numbers)”.  So there’s no difference from before what it means for $G_T$ to hold.  And there’s a difference, but a straightforward one, in whether we can show $G_T$ holds: If $T$ is a theory that we can prove consistent (so e.g. PA would be such a theory, if we’re working ambiently in something like ZFC), then using that, we can prove unconditionally that $G_T$ holds. If we can’t prove $T$ is consistent (e.g. if $T$ is ZFC itself, or something stronger), then all we can prove is: If $T$ is consistent, then $G_T$ holds.
A: There are several distinct issues to be aware of.
First, there is the question of the meaning of "true in a model" versus being simply "true" (or "true simpliciter" if you like Latin).  Once one realizes that it is possible to build different mathematical structures in which the Gödel sentence (or indeed any sentence in the formal language of arithmetic) can be interpreted in different ways, it might seem that, to avoid ambiguity, it is always necessary to specify which interpretation I have in mind when I say that the sentence is true or false. Technically there is some chance of confusion, but when it comes to sentences of arithmetic, it is always assumed that "true" without further qualification means "true when interpreted in terms of the standard integers."
The more subtle question is, when people study some arithmetic theory $T$ and say that the Gödel sentence $G$ for $T$ is "true," what justification do they have for claiming that $G$ is true?  At first glance, it might seem that such a claim is unwarranted.  Conventionally, in mathematics, we feel justified in confidently asserting that something is true if (and only if) we can prove it.  But if we're studying $T$, then we are typically interested in understanding what can be proved on the basis of $T$ itself.  $G$ is specifically constructed so as to be unprovable in $T$ (unless $T$ is inconsistent) so it seems particularly confusing to confidently assert that $G$ is true when we (apparently) can't prove $G$.
The first point to recognize is that when we're trying to decide whether we are justified in asserting that $G$ is true (equivalently, whether we can prove $G$), what is relevant is the metatheory in which we are working, rather than the theory $T$ itself.  When we're proving Gödel's theorem, we're reasoning about $T$, and whether we can prove this or that statement about $T$ depends on what metatheoretical principles we allow ourselves. The metatheory need not bear any particular relationship to $T$ itself, so the fact that $T$ does not prove $G$ does not automatically rule out the possibility that $G$ is provable in the metatheory.
Fine, you might say, but what if $T$ is something like PA, which also happens to be  a perfectly good system for performing metatheoretical arguments?  Can't we declare that our metatheory is also PA, and in that case, isn't it the case that we have no warrant for declaring (in the metatheory) that $G$ is true?
The answer is yes.  You might indeed be working in some metatheory which doesn't prove $G$.  Nothing in the proof of Gödel's theorem actually requires claiming that $G$ is true.  You are perfectly within your rights to work within some weak metatheory in which $G$ cannot be proved, meaning that you don't have any real justification for asserting that $G$ is true.
"But wait," you say, "now I'm more confused than ever.  I swear that I've read lots of accounts in which the truth of $G$ is asserted as fact.  Now you're telling me that such a claim isn't warranted?"  Well, I didn't quite say that.  I said that you might be working in some metatheory which doesn't prove $G$.  In many situations, though, you're studying a theory $T$ that hasn't been randomly plucked out of thin air; $T$ is being considered as a plausible candidate for the foundations of mathematics, and in particular, it is plausible that the theorems of $T$ are true (i.e., that $T$ is sound).  For example, if $T$ were obviously inconsistent, you wouldn't be interested in studying it, would you?  You also probably wouldn't consider $T$ to be a serious candidate for the foundations of mathematics if (for example) $T$ disproved Fermat's Last Theorem.  So in your metatheory, you probably want to assume—even though strictly speaking you don't have to—that $T$ is sound.  And from the assumption that $T$ is sound, we can easily prove $G$: if $G$ were false then $G$ would be provable in $T$, and the soundness of $T$ would imply that $G$ is true, which is a contradiction.  This is why accounts of Gödel's theorem—at least, those which emphasize the philosophical/metamathematical implications—typically say that $G$ is true.
A: In simple terms, it is true in this sense:
If you specify a logic system S1 by giving its axioms, there will be a Gödel sentence G1 that cant be decided within it. So the axioms cant be strong enough to prevent an undecidable proposition existing.
When you ask about adding an axiom A1 to make that sentence true or false, you are changing the axiomatic system we are discussing. You are now discussing S1+A1, call that system S2.
Its true that your original Gödel sentence G1 is now no longer undecidable in S2. Its clearly either true or false. But there will now exist a * different * Gödel sentence G2 that is undecidable in S2.
Gödels first incompleteness theorem states that however you try to strengthen S1, however many new axioms you bolster it with, its still incomplete. There will ALWAYS be a sentence Gn that is undecidable in your updated logic system Sn.
