The Continuum Hypothesis, Part II, Volume 48, Number 7

Introduction In the first part of this article, I identified the correct axioms for the structure 〈P(N),N,+, ·,∈〉 , which is the standard structure for Second Order Number Theory. The axioms, collectively “Projective Determinacy”, solve many of the otherwise unsolvable, classical problems of this structure. Actually working from the axioms of set theory, ZFC, I identified a natural progression of structures increasing in complexity: 〈H(ω),∈〉, 〈H(ω1),∈〉 , and 〈H(ω2),∈〉 , where for each cardinal κ, H(κ) denotes the set of all sets whose transitive closure has cardinality less than κ. The first of these structures is logically equivalent to 〈N,+, ·〉, the standard structure for number theory; the second is logically equivalent to the standard structure for Second Order Number Theory; and the third structure is where the answer to the Continuum Hypothesis, CH, lies. The main topic of Part I was the structure 〈H(ω1),∈〉 . Are there analogs of these axioms, say, some generalization of Projective Determinacy, for the structure 〈H(ω2),∈〉? Any reasonable generalization should settle the Continuum Hypothesis. An immediate consequence of Cohen’s method of forcing is that large cardinal axioms are not terribly useful in providing such a generalization. Indeed it was realized fairly soon after the discovery of forcing that essentially no large cardinal hypothesis can settle the Continuum Hypothesis. This was noted independently by Cohen and by Levy-Solovay. So the resolution of the theory of the structure 〈H(ω2),∈〉 could well be a far more difficult challenge than was the resolution of the theory of the structure 〈H(ω1),∈〉 . One example of the potential subtle aspects of the structure 〈H(ω2),∈〉 is given in the following theorem from 1991, the conclusion of which is in essence a property of the structure 〈H(ω2),∈〉 .