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1	The Existence of a Discontinuous Homomorphism Requires a Strong Axiom of Choice Andersen, Michael Steven 01 December 2014 (has links) (PDF) Conner and Spencer used ultrafilters to construct homomorphisms between fundamental groups that could not be induced by continuous functions between the underlying spaces. We use methods from Shelah and Pawlikowski to prove that Conner and Spencer could not have constructed these homomorphisms with a weak version of the Axiom of Choice. This led us to define and examine a class of pathological objects that cannot be constructed without a strong version of the Axiom of Choice, which we call the class of inscrutable objects. Objects that do not need a strong version of the Axiom of Choice are scrutable. We show that the scrutable homomorphisms from the fundamental group of a Peano continuum are exactly the homomorphisms induced by a continuous function.We suspect that any proposed theorem whose proof does not use a strong Axiom of Choice cannot have an inscrutable counterexample. inscrutable inscrutability scrutable scrutability axiom of choice discontinuous locally trivial non-locally trivial kernel invariance shelah pawlikowski countable choice choice arbitrary choice dependent choice discontinuity Mathematics