We isolate natural strengthenings of Bounded Martin’s Maximum which we call ${\mathsf{BMM}}^{*}$ and $A-{\mathsf{BMM}}^{*,++}$, and we investigate their consequences. We also show that if $A-{\mathsf{BMM}}^{*,++}$ holds true for every set of reals $A$ in $L$, then Woodin’s axiom $$ holds true. We conjecture that ${\mathsf{MM}}^{++}$ implies $A-{\mathsf{BMM}}^{*,++}$ for every $A$ which is universally Baire.

We show that L absoluteness for semi-proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L absoluteness for proper forcings. By [7], L absoluteness for stationary set preserving forcings gives an inner model with a strong cardinal. By [3], the Bounded Semi-Proper Forcing Axiom is equiconsistent with the Bounded Proper Forcing Axiom , which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martin's Maximum is much stronger than BSPFA in (...) that if BMM holds, then for every X ∈ V , X# exists. (shrink)

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM++ makes the Π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_2}$$\end{document}-fragment of the theory of Hℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\aleph_2}}$$\end{document} invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to (...) Hℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\aleph_2}}$$\end{document} of Woodin’s absoluteness results for L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document}. In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis. (shrink)

We show that under $\mathsf {BMM}$ and “there exists a Woodin cardinal, $"$ the nonstationary ideal on $\omega _1$ cannot be defined by a $\Pi _1$ formula with parameter $A \subset \omega _1$. We show that the same conclusion holds under the assumption of Woodin’s $(\ast )$ -axiom. We further show that there are universes where $\mathsf {BPFA}$ holds and $\text {NS}_{\omega _1}$ is $\Pi _1(\{\omega _1\})$ -definable. Lastly we show that if the canonical inner model with one Woodin cardinal (...) $M_1$ exists, there is a generic extension of $M_1$ in which $\text {NS}_{\omega _1}$ is saturated and $\Pi _1(\{ \omega _1\} )$ -definable, and $\mathsf {MA_{\omega _1}}$ holds. (shrink)

Several situations are presented in which there is an ordinal γ such that ${\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}$ is a stationary subset of ${[\gamma]^{\aleph_0}}$ for all stationary ${S, T\subseteq \omega_1}$ . A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of ${H_{\omega_2}}$ and the existence of sharps for all reals. Also, an (...) optimal model separating Bounded Semiproper Forcing Axiom (BSPFA) and Bounded Martin’s Maximum (BMM) is produced and it is shown that a strong form of BMM involving only parameters from ${H_{\omega_2}}$ implies that every function from ω 1 into ω 1 is bounded on a club by a canonical function. (shrink)

Nel 1966, quando questo libro venne pubblicato nella collana BMM di Mondadori, esso rispondeva all'esigenza di avere un'esposizione chiara e limpida dei concetti fondamentali della fenomenologia - indirizzo filosofico che si andava sempre più diffondendo in Italia. Di qui il suo disegno molto semplice, ma altrettanto preciso che, dopo una introduzione di carattere orientativo su Husserl e sul movimento fenomenologico, mette a fuoco le questioni di metodo e di contenuto che questo indirizzo solleva. Così vengono prese in considerazione le argomentazioni (...) scettiche che, approdando al dubbio cartesiano, aprono il tema dell'intenzionalità e della soggettività. Un ampio spazio viene dedicato alla fenomenologia della percezione ed al nuovo tema, emergente nella discussione con il marxismo, di una possibile fenomenologia del bisogno. A distanza di tanti anni, il libro non ha perso la sua vitalità, ancor più avvivata dalle notevoli integrazioni e aggiornamenti curati da Vincenzo Costa nel 2000. (shrink)