Papers by Giorgio Laguzzi
In this paper we analyse some questions concerning trees on κ, both for the countable and the unc... more In this paper we analyse some questions concerning trees on κ, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in [6, Question 5.2] about the diagram for regularity properties.
We provide a list of open problems in the research area of generalised Baire spaces, compiled wit... more We provide a list of open problems in the research area of generalised Baire spaces, compiled with the help of the participants of two workshops held in Amsterdam (2014) and Hamburg (2015).
We investigate some versions of amoeba for tree forcings in the generalized Cantor and Baire spac... more We investigate some versions of amoeba for tree forcings in the generalized Cantor and Baire spaces. This answers [9, Question 3.20] and generalizes a line of research that in the standard case has been studied in [10], [11] and [7]. Moreover, we also answer questions posed in [4] by Friedman, Khomskii and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show Σ 1 1-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between Ramsey and Baire property , in slight contrast with the standard case.
In this paper we introduce a tree-like forcing notion extending some properties of the random for... more In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of 2 κ , κ inaccessible, and study its associated ideal of null sets and notion of measurability. In particular , we answer a question of Shelah [10, Problem 0.4], about defining a forcing which is κ κ-bounding, < κ-closed and κ +-cc, for κ inaccessible. This also contributes to a line of research adressed in the survey paper [5].
We give a very brief survey on ZFC theory (Zermelo-Fraenkel Set Theory) and we present an intuiti... more We give a very brief survey on ZFC theory (Zermelo-Fraenkel Set Theory) and we present an intuitive introduction to the method of forcing and some applications to the real line. Our exposition will be very informal, without any claim of completeness and rigour. The idea is just to give a very intuitive idea of what Set Theory and forcing-method are, and why they are interesting and useful from the viewpoint of Mathematics. In the final part we also expose some results recently obtained, concerning regularity properties of the real numbers, such as Lebesgue measurability, Baire property, Ramsey property and Silver measurability.
We investigate two closely related partial orders of trees on ω ω : the full-splitting Miller tre... more We investigate two closely related partial orders of trees on ω ω : the full-splitting Miller trees and the infinitely often equal trees, as well as their corresponding σ-ideals. The former notion was considered by Newelski and Ros lanowski while the latter involves a correction of a result of Spinas. We consider some Marczewski-style regularity properties based on these trees, which turn out to be closely related to the property of Baire, and look at the dichotomies of Newelski-Ros lanowski and Spinas for higher projective pointclasses. We also provide some insight concerning a question of Fremlin whether one can add an infinitely often equal real without adding a Cohen real.
Mathematical Logic Quarterly, 2015
We present some results about the burgeoning research area concerning set theory of the "κ-reals"... more We present some results about the burgeoning research area concerning set theory of the "κ-reals". We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting.
Archive for Mathematical Logic, 2014
We present a model where ω1 is inaccessible by reals, Silver measurability holds for all sets but... more We present a model where ω1 is inaccessible by reals, Silver measurability holds for all sets but Miller and Lebesgue measurability fail for some sets. This contributes to a line of research started by Shelah in the 1980s and more recently continued by Schrittesser and Friedman (see ), regarding the separation of different notions of regularity properties of the real line.
Archive for Mathematical Logic, 2014
In this paper we analyse some notions of amoeba for tree forcings. In particular we define an amo... more In this paper we analyse some notions of amoeba for tree forcings. In particular we define an amoeba Sacks and prove that it satisfies the Laver property. Our construction will be different from the one presented by Shelah, Louveau and Velickovic in . Further we introduce an amoeba Silver and prove that it satisfies quasi pure decision but not pure decision. We finally present a generalized version of amoeba and discuss some interesting associated questions.
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Papers by Giorgio Laguzzi