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Within ZFC you can formalize Tarski's definition of truth, then prove that the axioms of ZFC PA are all true and that the rules of inference preserve truth. This gives a formal proof of Con(PA). This allows you to prove not just the consistency of PA, but the consistency of PA + Con(PA), or PA + Con(PA) + Con(PA+Con(PA)), etc. Nothing close to the full strength of ZFC is needed for any of this (though of course you need something beyond PA). |
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Within ZFC you can formalize Tarski's definition of truth, then prove that the axioms of ZFC are all true and that the rules of inference preserve truth. This gives a formal proof of Con(PA). This allows you to prove not just the consistency of PA, but the consistency of PA + Con(PA), or PA + Con(PA) + Con(PA+Con(PA)), etc. Nothing close to the full strength of ZFC is needed for any of this (though of course you need something beyond PA). |
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