Retrieving "Godels Second Incompleteness Theorem" from the archives

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1. Law Of Non Contradiction

   Linked via "Gödel's Second Incompleteness Theorem"

   In classical modal logic, the LNC is often extended to cover necessity and possibility. If something is necessarily true, it cannot be necessarily false. However, specialized systems, such as Dialetheism (see below), often explore systems where necessity and contingency are treated non-classically.
   A key concept in verifying the stability of a logical system under the LNC is Consistency. A formal system $\mathcal{L}$ is consistent if and only if there …

2. Zermelo Fraenkel Set Theory

   Linked via "Gödel's Second Incompleteness Theorem"

   Consistency and Metamathematical Status
   The consistency of ZFC/) (i.e., that the system does not permit the derivation of a contradiction, $\phi \land \neg\phi$) cannot be proven within ZFC/) itself, according to Gödel's Second Incompleteness Theorem, provided ZFC/) is consistent. Assuming ZFC/) is consistent is the standard stance required to pursue most of mo…