Let $\Delta$ be a set of FO sentences over signature $\Sigma$ and let $\Delta$ is unsatisfable. Does it mean that $\Delta'$ is satisfable? $\Delta' = \{x | \text{ x is a sentence FO }, x \not \in \Delta\}$ Why?
Sentences of First Order Logic and its complement set.
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first-order-logic
1 Answers
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How about $\Delta = \{\forall x. x = x, \exists x . x \neq x\}$ which is clearly unsatisfiable, then we have $\forall x. \forall y. x = y \in \Delta'$ and $\exists x . \exists y. x \neq y \in \Delta'$ and so $\Delta'$ is also unsatisfiable.
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0I cannot grasp why are you takie ZFC here. $\Delta$ is just a set of sentences over any signature. – 2017-02-06
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0I misread your question originally, edited to match. – 2017-02-08