  The Minimalist Foundation, for short MF, was conceived by the first author
with G. Sambin in 2005, and fully formalized in 2009, as a common core among
the most relevant constructive and classical foundations for mathematics.
  To better accomplish its minimality, MF was designed as a two-level type
theory, with an intensional level mTT, an extensional one emTT, and an
interpretation of the latter into the first.
  Here, we first show that the two levels of MF are indeed equiconsistent by
interpreting mTT into emTT. Then, we show that the classical extension emTT^c
is equiconsistent with emTT by suitably extending the G\"odel-Gentzen
double-negation translation of classical logic in the intuitionistic one. As a
consequence, MF turns out to be compatible with classical predicative
mathematics \`a la Weyl, contrary to the most relevant foundations for
constructive mathematics.
  Finally, we show that the chain of equiconsistency results for MF can be
straightforwardly extended to its impredicative version to deduce that
Coquand-Huet's Calculus of Constructions equipped with basic inductive types is
equiconsistent with its extensional and classical versions too.
