  This result is due to Orey and Hájek (for further details see Theorem 6 on p. 103 of Lindström (2003) and Fact 3.2 of Visser (1998)). It follows from this result and the second incompleteness theorem that for any theory T meeting these two conditions, the theory T + Con(T) is strictly stronger than T, that is, T &lt; T + Con(T). Moreover, it follows from the arithmetized completeness theorem that the T + ¬Con(T) is interpretable in T, hence, T ≡ T + ¬Con(T) (see Feferman 1960). 