Gödel and Adding Axioms I've been reading about Gödel's Incompleteness Theorems and there's something that I don't quite understand. It's about adding new statements as axioms to a system.

I'm not sure if I'm understanding anything wrong so here is a brief summary of what I think I know:

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*Gödel's First Incompleteness Theorem

Any consistent system $S$ strong enough to express arithmetic is incomplete. Gödel shows this by explicitly constructing an undecidable statement G that's reads something like:
G $\iff$ There is no Gödel number $x$ that corresponds to a proof of G in $S$
Assuming $S$ is consistent, if G is provable, then there exists $x$ that proves G, which is a contradiction. If not G is provable, then there is a proof of G, which leads to a contradiction. Therefore, G cannot be proven or disproven, so $S$ is incomplete.


*Gödel's Second Incompleteness Theorem

If $S$ is proven to be consistent, then $S$ is inconsistent. We get this by noting the implication 'if $S$ is consistent, then G is true'. So if there is a proof of $S$'s consistency, then we can prove $G$, which leads to a contradiction.

I understand that if a statement $A$ is independent of a set of consistent axioms, then there are models where $A$ or not $A$ are true, and I can add $A$ or not $A$ to the axioms while maintaining consistency.
The problem is that '$S$ is consistent' is undecidable, so does that mean I can add '$S$ is inconsistent' as an axiom? What are the implications of adding such an axiom? I understand that adding this as an axiom won't lead to contradictions, because I can't prove '$S$ is consistent' anyway, but I just find something very strange about adding this as an axiom.
 A: Your understanding is correct, apart from the caveat that there are a lot of details you glossed over. Most significantly, in the part about the second incompleteness theorem, while the first incompleteness theorem shows, 'if S is consistent, then G is true', what we really need is to show that S can prove this... and this is a major undertaking, even on top of the detailed work to show the first incompleteness theorem, and it requires new technical assumptions about S, more detailed than the conditions needed for the first theorem.
But let's put that aside, and address your question. Yes, it certainly feels odd that it is consistent to add "S is inconsistent" as an axiom (sometimes I think that this should be a "paradox" with a name). The reason it feels odd is that you are adding an axiom that is false.
The crucial thing to grasp here is that a theory doesn't need to be correct (according to some standard interpretation) in order to be consistent. In terms of interpretations, the standard interpretation (i.e. the natural numbers) is no longer a model of "S + S is inconsistent", but some of the nonstandard interpretations of S are still models. These nonstandard models have nonstandard natural numbers (larger than any number their initial segment that looks like $\mathbb N$) that they think encode proofs of "0=1" in S.
In "S + S is inconsistent", we also have an odd syntactic situation where for each natural number $n,$ it is provable that "$\mathbf n$ does not code a proof of 0=1 in S" and yet it is also provable that "there is a proof of 0=1 in S". This is called $\omega$-inconsistency. Of course, as in the previous paragraph, we can see that the resolution is that not every element of the domain of an interpretation needs to be represented by a numeral $\mathbf n.$
Perhaps we would like it if it were inconsistent to be wrong, but this would require that our theory decide every sentence (correctly), since, as you've reasoned, if a sentence isn't decidable, you can add either it or its negation and get a consistent system, and both can't be right. And unfortunately this is exactly what Godel's theorem rules out (for sufficiently strong, recursively axiomatizable theories).
