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ANALYSIS IN J2

This is an expository paper in which I explain how core mathematics , particularly abstract analysis, can be developed within a concrete countable set J 2 (the second set in Jensen's constructible hierarchy). The implication , well-known to proof theorists but probably not to most mainstream mathematicians, is that ordinary mathematical practice does not require an enigmatic metaphysical universe of sets. I go further and argue that J 2 is a superior setting for normal mathematics because it is free of irrelevant set-theoretic pathologies and permits stronger formulations of existence results. Perhaps many mathematicians would admit to harboring some feelings of discomfort about the ethereal quality of Cantorian set theory. Yet draconian alternatives such as intuitionism, which holds that simple number-theoretic statements like the twin primes conjecture may have no definite truth value, probably violate the typical working mathematician's intuition far more severely than any vague unease he may feel about remote cardinals such as, say, ℵ ℵω. I believe that ordinary mathematical practice is actually most compatible with an intermediate foundational stance I call conceptualism. This is a modernized version of the predicativist philosophy originally formulated by Henri Poincaré and Bertrand Russell, which treats elementary number theory from an essentially pla-tonic point of view but in the realm of set theory admits only those sets that can be constructed in a very concrete way. One consequence of this restriction is that on the conceptualist account all sets are countable. While this would seem to render the bulk of ordinary mathematics illegitimate, work begun by Hermann Weyl and continued by many others has gradually shown that in fact most if not all core mathematics can be developed within surprisingly minimalist systems. This point has been emphasized in [3]. However, early work of this type tended to be presented in a somewhat abstruse or idiosyncratic way, while more recent work is generally couched in axiomatic frameworks that mainstream mathematicians might find un-intuitive. The goal of this paper is to show how to develop core mathematics in a concrete countable domain called J 2 which should be relatively easily appreciated by mainstream mathematicians with no special training in logic. (The first two sections of the paper are probably the main hurdle; however, they might not need to be read in detail before proceeding to later sections.) Our central novelty is not any major technical advance but rather the choice of an approach …