Measurable Cardinals and Constructible Sets

A cardinal number m will be called measurable if and only if there is a set X of cardinality m and a non-trivial, real-valued, countably additive measure μ defined on all subsets of X. (The term non-trivial can be taken to mean that μ(X) = 1 and μ({x}) = 0, for all χ ɛ X). If 2^ℵ₀ = ℵ₁, Banach and Kuratowski [1] proved that ℵ₁ is not measurable. Ulam [12] proved that if there is a measurable cardinal, then either 2^ℵ₀ is measurable or there exists a 2-valued measurable cardinal (2-valued in the sense that the measure μ can be assumed to take on only the values 0 and 1). Ulam and Tarski showed that no cardinal less than the first strengly inaccessible cardinal beyond ℵ₀ can be 2-valued measurable (cf. [12], esp. footnote 1, p. 146). Last year, using some new results of Hanf, Tarski proved [11] that many inaccessibles, in particular the first beyend ℵ₀, are not 2-valued measurable (for other proofs cf. [6] and [2]). Even though the least 2-valued measurable cardinal, if it exists at all, now appears to be incredibly large since Tarski's results apply to a seemingly inexhaustible number of inaccessible cardinals, it still seems plausible to many people including the author to assume that such cardinals do exist. However, this assumption has some surprising consequences, for, as shall be outlined below, we can show that the existence of measurable cardinals contradiets Gödel's axiom of constructibility…