Diaconescu showed that the Axiom of Choice entails Excluded-Middle
in topoi
Diaconescu75. Lacas and Werner adapted the proof to show
that the axiom of choice in equivalence classes entails
Excluded-Middle in Type Theory
LacasWerner99.
Three variants of Diaconescu's result in type theory are shown below.
A. A proof that the relational form of the Axiom of Choice +
Extensionality for Predicates entails Excluded-Middle (by Hugo
Herbelin)
B. A proof that the relational form of the Axiom of Choice + Proof
Irrelevance entails Excluded-Middle for Equality Statements (by
Benjamin Werner)
C. A proof that extensional Hilbert epsilon's description operator
entails excluded-middle (taken from Bell
Bell93)
See also
Carlström for a discussion of the connection between the
Extensional Axiom of Choice and Excluded-Middle
Diaconescu75 Radu Diaconescu, Axiom of Choice and Complementation,
in Proceedings of AMS, vol 51, pp 176-178, 1975.
LacasWerner99 Samuel Lacas, Benjamin Werner, Which Choices imply
the excluded middle?, preprint, 1999.
Bell93 John L. Bell, Hilbert's epsilon operator and classical
logic, Journal of Philosophical Logic, 22: 1-18, 1993
Carlström04 Jesper Carlström, EM + Ext + AC_int <-> AC_ext,
Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004.
Pred. Ext. + Rel. Axiom of Choice -> Excluded-Middle
From predicate extensionality we get propositional extensionality
hence proof-irrelevance
From proof-irrelevance and relational choice, we get guarded
relational choice
The form of choice we need: there is a functional relation which chooses
an element in any non empty subset of bool
Thanks to the axiom of choice, the boolean witnesses move from the
propositional world to the relevant world
An alternative more concise proof can be done by directly using
the guarded relational choice