Related papers: Forcing and classes of $\mathsf{HYP}$-dominating f…
In this article we adapt the existing account of class-forcing over a ZFC model to a model $(M,\mathcal{C})$ of Morse-Kelley class theory. We give a rigorous definition of class-forcing in such a model and show that the Definability Lemma…
A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is…
$\Sigma^1_3$-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…
Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the…
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…
We present a notion of forcing that can be used, in conjunction with other results, to show that there is a Martin-L\"of random set X such that X does not compute 0' and X computes every K-trivial set.
We make use of some observations on the core model, for example assuming $V=L [ E ]$, and that there is no inner model with a Woodin cardinal, and $M$ is an inner model with the same cardinals as $V$, then $V=M$. We conclude in this latter…
Given a countable Turing ideal $\mathcal{I} \subseteq \omega^{\omega}$, we say that $x$ is a list (resp. weak list) of $\mathcal{I}$ if $\mathcal{I}=\{x^{[n]} : n \in \omega\}$ (resp. if $\mathcal{I} \subseteq \{x^{[n]} :n \in \omega\}$).…
We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(chi), in). This leads to forcing notions which are…
For any system with limited statistical knowledge, the combination of evidence and the interpretation of sampling information require the determination of the right reference class (or of an adequate one). The present note (1) discusses the…
We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we…
We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of…
This is an overview about a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along $\omega_1$ is given. Then its direct limit satisfies ccc by…
We investigate how set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory $\langle M,\in^M\rangle$, we explain senses in which one may compute…
We show that it is possible to add $\kappa^+-$Cohen subsets to $\kappa$ with a Prikry forcing over $\kappa$. This answers a question from \cite{HayutBenhanouGitik}. A strengthening of non-Galvin property is introduced. It is shown to be…
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are…
This work is a part of my upcoming thesis [7]. We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2)…
Taking a compact K\"{a}hler manifold as playground, we explore the powerfulness of Hodge index theorem. A main object is the Lorentzian classes on a compact K\"{a}hler manifold, behind which the characterization via Lorentzian polynomials…
We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the…
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing…