We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah’s iteration theorems appearing in Chapters V and VIII of Proper an...

We consider the question, which of the major classes defined by topological diagonalizations of open or Borel covers is provably, or at least consistently, hereditary. Many of the classes in the open case are not hereditary already in ZFC. We show that none of them is provably hereditary. This is contrasted with the Borel case, where some of the classes are provably hereditary. We also give two...

Set theory entered the modern era through the work of Gödel and Cohen. This work provided set-theorists with the necessary tools to analyse a large number of mathematical problems which are unsolvable using only the traditional axiom system ZFC for set theory. Through these methods, together with their subsequent generalisation into the context of large cardinals, settheorists have had great su...

An axiomatic development in ProofPower-HOL of a higher order theory of well-founded sets. This is similar to a higher order ZFC strengthened by the assertion that every set is a member of some other set which is a (standard) model of ZFC. Created 2007/09/25 Last Change Date: 2012-08-11 21:01:53 http://www.rbjones.com/rbjpub/pp/doc/t023.pdf Id: t023.doc,v 1.17 2012-08-11 21:01:53 rbj Exp c © Rog...

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in ZFC and even in the theory ZFC + c = ω2 if the number of pieces can be uncountable but less than the continuum. We also ...

The consistency of the theory ZF+AD+“every set of reals is universally Baire” is proved relative to ZFC + “there is a cardinal λ that is a limit of Woodin cardinals and of strong cardinals.” The proof is based on the derived model construction, which was used by Woodin to show that the theory ZF+AD+“every set of reals is Suslin” is consistent relative to ZFC+“there is a cardinal λ that is a lim...

We present some results concerning extensions of models of ZFC in which cofinalities of cardinals are changed and/or cardinals are collapsed, in particular on minimal such extensions. Our main tools are perfect tree forcing PF(S) and Namba forcing Nm(S). We prove that if N 2 M is an extension such that (i) M k K = I+ > H,, (ii) 1 fl N E M and (iii) N = Mlf] for some cofinal f : oo+ K, then N 2 ...

We prove the following characterizations of nonstandard models ZFC (Zermelo-Fraenkel set theory with axiom choice) that have an expansion to a model GB (Gödel-Bernays class theory) plus Δ11-CA (the scheme Δ11-Comprehension). In what follows, M(α):=(V(α),∈)M, LM is formulae infinitary logic L∞,ω appear in well-founded part M, and Σ11-AC Σ11-Choice. Theorem A. The are equivalent for M any cardina...

We prove that if cf(λ) > א0 and 2 cf(λ) < λ then λ → (λ, ω + 1) in ZFC