Is Godel's system of axioms inconsistent?

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The discussion centers on the consistency of Gödel's system of axioms, specifically Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). It is established that Gödel's incompleteness theorem indicates that any consistent mathematical theory that includes a model of natural numbers is inherently incomplete. The conversation highlights that while we cannot prove ZFC's consistency, it does not imply that ZFC is inconsistent. Instead, if a statement is unprovable in one consistent system, it can be provable in another consistent system.

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As we cannot prove that Godel's system of axioms (ZFC?) is consistent, is it possible that it is inconsistent, that the Godel sentence is false, and that we yet prove it to be 'true'?
 
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When using an inconsistent system, any statement can be proven true.

If a statement is unprovable in one consistent system, then there is a consistent system in which this statement is provable, and there is another consistent system in which this statement is disprovable.
 
I.e., yes.

Godel's theorem says that "any consistent mathematical theory containing a model of the natural numbers is incomplete". (The conditions on the theory are a bit stronger than that, but the consistent requirement is part of the statement.)

If you follow his proof, you can see exactly where he assumes consistency of the theory, though it has been long enough that I have forgotten.
 
I may be wrong, but I don' t think it's a consequence of Godel that we can't prove ZFC consistent. We can prove all kinds of things to be consistent. What Godel says is that if we can write a formula in the language of ZFC that says "ZFC is consistent", then we can't prove it unless ZFC is really inconsistent.
 

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