We show that the semi-linear ordering principle for continuous functions implies the determinacy of all Wadge and Lipschitz games.

We prove that the determinacy of all Lipschitz games, the determinacy of all Wadge games, and the semi-linear ordering principle for Lipschitz maps are all equivalent.

Assuming AD + DC, we characterize the self-dual boldface pointclasses which are strictly larger than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank [Formula: see text], and those of Wadge rank [Formula: see text] when ξ is limit.

It is shown, assuming the existence of a Woodin cardinal δ, that every tree ordering on some limit ordinal $\lambda < \delta$ with a cofinal branch is the tree ordering of some iteration tree on V.

We introduce two new iteration games: the game , which is a strengthening of the weak iteration game, and the game , which is somewhat stronger than but weaker than the full iteration game of length ω1. For a countable M elementarily embeddable in some Vη, with two players I and II, we can show that II wins and that I does not win.

In this paper we continue the line of work initiated in “Building iteration trees”. It is shown that the existence of a certain kind of iteration tree of height ω is equivalent to the existence of a cardinal δ that is Woodin with respect to functions in the next admissible.

In this paper we introduce a new hierarchy of large cardinals between \ and \, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.

In this paper we introduce a new hierarchy of large cardinals between I3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{\mathsf {I3}}}}$$\end{document} and I2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{\mathsf {I2}}}}$$\end{document}, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.