It is probably commonplace today to think of mathematical theories as ordered by “strength.” Most common is probably the ordering by relative interpretability, which is a partial ordering modulo mutual relative interpretability. However, Paul Bernays, one of the first to think of theories in this way, put it in terms of the strength of platonist assumptions, so that, for example, classical arithmetic is higher than intuitionistic arithmetic, although they are mutually interpretable.
I discuss such hierarchies in relation to problems of evidence, since one would expect stronger theories to be less evident than weaker ones. In set theory theories are formulated that are very high in this hierarchy. We consider in this light some problems of evidence in set theory.
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Philosophy of Mathematics Seminar Convenors: Daniel Isaacson, Volker Halbach and James Studd