Heterologicality and imcompleteness (Q2776814)

For a theory \(T\) containing ZF, the author presents a semantic proof of the following: if \(T\) has a model, then \(T+\neg \text{Con}_T\) has a model. This is a version of Gödel's Second Incompleteness Theorem. The author remarks it is equivalent to the usual version, provided that one has the derivability conditions characterizing provability in \(T\), otherwise it is weaker. Motivated by Grelling's antinomy, the author produces a sentence \(\text{HET}_T\) that can be interpreted as saying the predicate ``heterological'' is itself heterological. He then shows that \(\text{HET}_T\) does not follow from \(T\) and that it is provably equivalent in \(T\) to the consistency of \(T\). The author indicates how to give a similar result for Peano Arithmetic.