Related papers: The Bounded Proper Forcing Axiom

There exists a family $\{B_{\alpha}\}_{\alpha<\omega_1}$ of sets of countable ordinals such that o $\max B_{\alpha}=\alpha$, o if $\alpha\in B_{\beta}$ then $B_{\alpha}\subseteq B_{\beta}$, o if $\lambda\leq \alpha$ and $\lambda$ is a limit…

We prove forcing axiom equivalents of two families of weakenings of the axiom of choice: a trichotomy principle for cardinals isolated by L\'evy, ${\rm H\hskip0.05pt}_\kappa$, and ${\rm DC}_\kappa$, the principle of dependent choices…

$\Sigma^1_3$-absoluteness for ccc forcing means that for any ccc forcing $P$, ${H_{\omega_1}}^V \prec_{\Sigma_2}{H_{\omega_1}}^{V^P}$. "$\omega_1$ inaccessible to reals" means that for any real $r$, ${\omega_1}^{L[r]}<\omega_1$. To measure…

I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma$: the hierarchy of bounded forcing axioms, of $\Sigma^1_1$-absoluteness and of Aronszajn tree preservation principles. The latter…

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a…

Suppose that $T^*$ is an $\omega_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong…

We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal $\kappa$ such that $2^\kappa > kappa^+$ then there…

A narrow system is a combinatorial object introduced by Magidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinal $\kappa$ satisfies the \emph{narrow…

We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega. In section 2 we show…

We continue the development of the theory of construction schemes over $\omega_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use…

It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if…

We use forcing over admissible sets to show that, for every ordinal $\alpha$ in a club $C\subset\omega_1$, there are copies of $\alpha$ such that the isomorphism between them is not computable in the join of the complete $\Pi^1_1$ set…

Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We…

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma_1$-definability at uncountable regular cardinals. In particular we give its exact consistency strength firstly in terms of the second…

We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $\omega_1$ can not be defined by a $\Sigma_1$ formula with parameter $A \subset \omega_1$. We show that the same conclusion holds under the…

In the absence of the Axiom of Choice, the "small" cardinal $\omega_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say…

We show that, assuming GCH, if $\kappa$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and obeying the constraints of Easton's theorem, namely,…

We show that it is consistent from an inaccessible cardinal that classical Namba forcing has the weak $\omega_1$-approximation property. In fact, this is the case if $\aleph_1$-preserving forcings do not add cofinal branches to…

The weakly compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact…

Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $\kappa$ is a singular strong limit of uncountable cofinality, all scales on $\kappa$ are…