Tim Gowers has a very nice piece on his blog about functions, multivalued functions, relations and the like, called “Why aren’t all functions well-defined?“.

Tim Gowers has a very nice piece on his blog about functions, multivalued functions, relations and the like, called “Why aren’t all functions well-defined?“.

Aatu KoskensiltaThe intuitionists have it easy, of course: they can just waffle about

extensionaloperations. (One is reminded of the trivial but fallacious intuitionistic proof of (certain form of) choice…)A standard illustration of the hazards of talking about well-definedness of functions is provided by the proof that recursive definitions (along some well-ordering, say) define unique functions. (At least I seem to recall this is a standard illustration.) Given a recursive definition we prove by induction, so one naturally surmises, that the function is well-defined. This is poppycock, as a moment's reflection reveals: we're trying to apply induction to show that an object we haven't proven to exist has a property that we haven't given any mathematical definition. And so it goes.