Related papers: Measuring club sequences, together with the Contin…
In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation,…
Let $X$ and $Y$ be topological spaces, let $Z$ be a metric space, and let $f: X\times Y\to Z$ be a mapping. It is shown that when $Y$ has a countable base $\mathcal B$, then under a rather general condition on the set-valued mappings $X\ni…
The purpose of this paper is two-fold. First, we would like to get rid of common assumption that causal set is bounded and attempt to model its scalar field action under the assumption that it isn't. Secondly, we would like to propose…
Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting…
The study of inner models was initiated by G\"odel's analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by…
We generically construct a model in which the ${\Pi^1_3}$-uniformization property is true, thus lowering the best known consistency strength from the existence of $M_1^{\#}$ to just $\mathsf{ZFC}$. The forcing construction can be adapted to…
We construct models, by three-dimensional arrays of ccc posets, where many classical cardinal characteristics of the continuum are pairwise different.
Consider $(\kappa^{+++},\kappa^{++}) \twoheadrightarrow (\kappa^+,\kappa)$ where $\kappa$ is an uncountable regular cardinal. By a result of Shelah's we have $\operatorname{cof}(X \cap \kappa^{++}) = \kappa$ for almost all $X \subset…
A club consisting of former regulars is added to an inaccessible cardinal, without changing cofinalities outside it. The initial assumption is optimal. A variation of the Radin forcing without a top measurable cardinal is introduced for…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vop\v{e}nka's…
We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not…
Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa} = \kappa$ or just ${\rm cf}(\kappa) > \omega$, $2^{2^{<\kappa}}= 2^\kappa$, and $([\kappa]^\kappa, \supseteq)$ collapses $2^\kappa$ to $\omega$. We show under these assumptions…
G\"odel's first and second incompleteness theorems are corner stones of modern mathematics. In this article we present a new proof of these theorems for ZFC and theories containing ZFC, using Chaitin's incompleteness theorem and a very…
We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different,…
We discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by Lopez-Ruiz, Mancini and Calbet (LMC) [Phys. Lett. A 209 (1995) 321]. In general, the extension of a magnitude from the…
We give a direct, detailed and relatively short proof of Shelah's theorem on club guessing sequences on $S^{\mu^+}_\mu$ (for any regular, uncountable cardinal $\mu$).
We develop a general framework (multidimensional asymptotic classes, or m.a.c.s) for handling classes of finite first order structures with a strong uniformity condition on cardinalities of definable sets: The condition asserts that…
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a…
We show that the consistency of $\mathrm{ZF} + \mathrm{AD}_{\mathbb{R}} + ``\Theta$ is measurable$"$ implies the consistency of $\mathrm{ZF} +``\Theta$ is the least strongly regular cardinal and the least measurable cardinal$"$ + $``$all…
Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\Sigma$ of $\mathcal{L}_{\omega_1, \omega}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection…