A forcing poset of size 221 which adds no new reals is described and shown to provide a Δ22 definable well-order of the reals . The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.

We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...) other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory. (shrink)

In [P. Larson, Martin’s Maximum and the axiom , Ann. Pure App. Logic 106 135–149], we modified a coding device from [W.H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co, Berlin, 1999] and the consistency proof of Martin’s Maximum from [M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annal. Math. 127 1–47] to show that from a supercompact limit of supercompact cardinals one could force Martin’s (...) Maximum to hold while the axiom fails. Here we modify that argument to prove a stronger fact, that Martin’s Maximum is consistent with the existence of a wellordering of the reals definable in H without parameters, from the same large cardinal hypothesis. In doing so we give a much simpler proof of the original result. (shrink)

This book brings together several directions of work in model theory between the late 1950s and early 1980s.