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
[Carlström04] 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 is equivalent
to AC_ext, Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004.
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