Related papers: Gap Forcing
A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method…
We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $\kappa$, if $\kappa$ is…
We show that every Jonsson cardinal is Ramsey in the Steel core model, provided that this model exists and there is no model with a Woodin cardinal. This basic result is improved in two directions. First, we prove the same result for…
In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very…
We study gauge coupling unification in the presence of extra dimensions compactified at a few TeV. Achieving unification requires a large number of gauge boson Kaluza-Klein excitations lighter than the string scale, such that the…
We introduce the forcing property of descending distributivity. A forcing $\mathbb{P}$ is $\kappa$-descending distributive if for all decreasing sequences $(D_\alpha)_{\alpha<\kappa}$ of open dense sets, $\bigcap_\alpha D_\alpha$ is open…
Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as…
For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and…
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…
The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for $\aleph^{\aleph_0}_0$; in spite of this similarity, the Cohen forcing and Random…
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…
A decay of weakly metastable phase coupled to two-dimensional Liouville gravity is considered in the semiclassical approximation. The process is governed by the ``critical swelling'', where the droplet fluctuation favors a gravitational…
We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located…
In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence phi holding in some forcing extension V^P and all subsequent extensions V^P*Q holds…
Assume V=L and lambda is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext(G,Z) (under well known limitations), we construct an abelian group G of…
Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…
The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver's theorem and Bukovsk\'y's theorem assert that set-generic extensions of a given…
Recently, a scenario has been proposed in which the gravitational scale could be as low as the TeV scale, and extra dimensions could be large and detectable at the electroweak scale. Although supersymmetry is not a requirement of this…
We study the constraints on gravity scale $M_P$ in extra-dimension gravitational theory, obtained from gravity-induced processes. The obtained constraints are subdivided into strong (though not robust) and reliable (though less strong). The…
Assume ZF($j$) and there is a Reinhardt cardinal, as witnessed by the elementary embedding $j:V\to V$. We investigate the linear iterates $(N_{\alpha},j_{\alpha})$ of $(V,j)$, and their relationship to $(V,j)$, forcing and definability,…