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It is my understanding that functional extensionality in Lean itself follows from the axiom propositional extensionality, so in that sense LEM is still a conseq
by tlringer 6y ago
It is my understanding that functional extensionality in Lean itself follows from the axiom propositional extensionality, so in that sense LEM is still a consequence of an axiom. The core theory of Lean is constructive.
- kmill 6y agoI'm not an expert on the foundations of Lean, but I do know that funext in the source code follows only from the axioms of quotient types and the eliminator for equality. There are a number of axioms that are implicit in the type checker itself, so you might say that proof irrelevance is built-in in that way. Propositional extensionality (propext) is that propositions that imply each other are equal, and that's an axiom in the library, but funext does not depend on propext.
- kmill 6y agoCorrecting my non-expert inaccuracies in this thread: I just learned that Diaconescu's theorem depends on the axiom of choice, which I'd missed even though I'd glanced at the code for classical.em a few times. So, LEM definitely depends on the axiom of choice in Lean.