Ralph Loader writes:
I'd much rather that my formalisation of mathematics could state nonsensical things (consistency strength isn't an issue here), than to be living in the fear that my formalisation may be inadequate to express some sensible arguments.
I for one _do_ live in fear (not really ... more a sort of smug schadenfreude) that set theoretic formalisation is inadequate to express some sensible arguments. The nonsensical things in set theory seem to arise from a basic nonsensical postulate: that there is a single fundamental relation "element of" on the mathematical universe that is sufficient to determine the nature of every mathematical object. It's a tour de force of modern mathematics that so much has been accomplished using this postulate, but its essential unreasonableness should not be forgotten and it is therefore complacent to assume that it is adequate to express all sensible arguments. Let me give some examples. 1) The "element of" relation is absolutely _un_fundamental - this is part of the force of Freyd's example about simple groups. What are the elements of a real number? If you consider the real number to be a Dedekind section, then it is a pair of subsets of the rationals, Q, and hence its elements are whatever you think the elements of an ordered pair are. Or, equivalently, it can be represented as a subset of Qx2 (or Q x any doubleton {L,R}). Or, equivalently, a subset of QxQ, with (q,r) in x iff |q-x| < r (i.e. the real is identified with its neighbourhood filter of rational open balls). All these enable a real to be described as a set, but they give it completely different elements. In reality there is no single universal "element of" relation that describes the nature of everything, including the reals; instead, the reals are described by various relations with other specific objects. 2) Topology: Normally one thinks of open sets as being sets of points, but localic topology views points as being sets of opens (e.g. reals as neighbourhood filters above). There is obviously a fundamental relation of points being "in" opens, and localic arguments can be expressed quite reasonably using it. Set theoretic expression using "element of" completely obscures this. In other words, set theory prevents you from adequately expressing reasonable arguments. 3) Generic objects: Suppose we agree on a particular set theoretic representation, e.g. reals as Dedekind sections. What then is a _generic_ real, such as the x in f(x) = x^2? Set theory has to treat this as a mere hole where a specific real could be put, but it is quite reasonable to treat it as a real in its own right (so long as you don't use - e.g. - excluded middle to demand specific properties of it) and that's exactly what people do. What are its elements? One can still make sense of the idea that it is determined by its elements (using "generalized elements" and Kripke-Joyal semantics), but in doing so one must go far beyond classical set theory. 4) What about theories such as that of accessible categories, that, for set theoretic reasons, have to be liberally sprinkled with infinite cardinals? Doesn't this make you think that perhaps set theory is somehow obscuring simple ideas? To my mind, the evidence suggests that despite its undoubted successes, set theory is not right for mathematical foundations, and we should be looking for its replacement. It is easy to be bemused by the fact that all our mathematical upbringing presumes set theoretical foundations, but we should try to recognize its limitations and failures. Steve Vickers.