Summary
Since P.J. Cohen's proof of the independence of the continuum hypothesis from the other axioms of set theory, there has been a remarkable flowering of similar results, ranging over all those branches of mathematics which deal in propositions of a similar level. Many of these are specific constructions dealing with individual problems. But some of the alternative models of set theory that have been developed provide answers to several questions. In terms of the number and variety of their uses, two are at present outstanding: model Δ of Gödel 40, and the models of Solovay & Tennenbaum 71. Each of these was constructed with the aim of showing the consistency of a particular hypothesis (in the former, the continuum hypothesis; in the latter, Souslin's hypothesis); but in each case an enormous number of unexpected further properties has emerged.
The structure of model Δ is such that, although it can be regarded as ordinary mathematics with one extra axiom added (the axiom of constructibility, or ‘V = L’), it is not possible to make deductions from this axiom without appealing to non-trivial ideas from mathematical logic; so that the non-logician who wishes to examine its consequences must work from one level lower (e.g. from R.B. Jensen's principle ♦). But the most useful properties of the Solovay–Tennenbaum model (or, rather, models) are relatively accessible, being derived by conventional arguments from Martin's axiom, ‘MA’, ‘m = c’, in one of its variations.
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- Consequences of Martin's Axiom , pp. ix - xiiPublisher: Cambridge University PressPrint publication year: 1984