We give a construction of the square principle by means of forcing with finite conditions.

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

We develop a general framework for forcing with coherent adequate sets on [Formula: see text] as side conditions, where [Formula: see text] is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of [Formula: see text] with finite conditions while preserving CH, solving (...) a problem of Friedman [Forcing with finite conditions, in Set Theory: Centre de Recerca Matemática, Barcelona, 2003–2004, Trends in Mathematics, pp. 285–295.]. (shrink)

We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than $\aleph_{1}$, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important (...) since a proof using finite supports is more amenable to generalizations to cardinals greater than $\aleph_{1}$. (shrink)

We investigate how weak square principles are denied by Chang’s Conjecture and its generalizations. Among other things we prove that Chang’s Conjecture does not imply the failure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}, i.e. Chang’s Conjecture is consistent with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}.