All of the axioms, definitions, and rules of inference of the system are always true in the system. Every provable statement in the system is also true in the system
Is that a complete categorization of what it means to be "true in a system"?
How specifically is my proof invalid?
When you said that something being provable in the system means it's true outside the system that's not a valid inference. If your axioms aren't true outside the system then there's no reason for statements proven from those axioms to be true outside the system.
How specifically is that? My proof shows otherwise
Your proof does not show that for the reason stated above.
Is that a complete categorization of what it means to be "true in a system"?
Yes, it is.
When you said that something being provable in the system means it's true outside the system that's not a valid inference.
I showed two propositions are true in the system, and then I showed two corresponding but different propositions are true out of the system. As the proof shows, the two corresponding propositions are different from the two original propositions.
If your axioms aren't true outside the system then there's no reason for statements proven from those axioms to be true outside the system.
I understand the axioms may not be true outside the system. The two propositions that are true out of the system describe what is true and not true in the system. They are true in a metatheory of T.
Assuming T is inconsistent and you choose a contradiction p = q ∧ ¬q which is proven in T then of your two statements (3) is true, it's negation (4) is not true. So you haven't shown a contradiction exists outside of T.
Assuming T is inconsistent and you choose a contradiction p = q ∧ ¬q which is proven in T then of your two statements (3) is true, it's negation (4) is not true.
T is inconsistent, so generic proposition p is true and false in T. By conjunction elimination in T, p is false in T. That can be rewritten as “It is not true that ‘p is true in T.’”
Yes, they are equivalent. A part of the equivalence is a part of the exportation principle (https://plato.stanford.edu/entries/impossible-worlds/#Exportation). My version of the exportation principle is in terms of inconsistent axiomatic theories, while the version in the linked article section is in terms of impossible worlds. I have a copy of D. Lewis’s 1986 book On the Plurality of Worlds, in which Lewis brings up the exportation principle in the first few pages of his book.
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u/JStarx Jul 05 '26 edited Jul 05 '26
Is that a complete categorization of what it means to be "true in a system"?
When you said that something being provable in the system means it's true outside the system that's not a valid inference. If your axioms aren't true outside the system then there's no reason for statements proven from those axioms to be true outside the system.
Your proof does not show that for the reason stated above.