1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The back-and-forth construction 3

Abstract not found

Abstract. This is an expository paper in which I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set J2 (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 J2 is a superior setting for normal mathematics because it is free of irrelevant settheoretic 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, ℵℵω. I believe that ordinary mathematical practice is actually most compatible with