A subset X of the Cantor space, 2^ω, is a lambda-prime-set iff for every countable subset Y of the Cantor space Y is relatively G-delta in X union Y. In this paper we prove two forcing results about lambda-prime-sets. First we show that it is consistent that every lambda-prime-set is a gamma-set. Secondly we show that is independent whether or not every dagger-lambda-set is a lambda-prime-set.
Let ${\bf x}=(x_n)_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\|\sum_{n\in F}x_n\|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B({\bf x})$ iff there is a non pathological lower semicontinuous submeasure $φ$ on $\mathbb{N}$ such that $\mathcal{I} =FIN(φ)=\{A\subseteq \mathbb{N}: \;φ(A)<\infty\}$. We address the questions of when $FIN(φ)$ is a tall ideal and has a Borel selector. W…
For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134
We say that X x Y satisfies the Uniquely Universal property (UU) iff there exists a set U open in X x Y such that for every open set W in Y there is a unique cross section U_x of U with U_x=W. Michael Hrusak raised the question of when does X x Y satisfy UU and noted that if Y is compact then X must have an isolated point. We prove the following: 1. If Y is a locally compact noncompact Polish space, then C x Y has UU where C is the Cantor space. 2. If Y is Polish, then B x Y has UU iff Y is not compact where B is the Baire space. 3. If Y is a sigma-compact subset of a Polish space which is not…
We give a semantics for the lambda-calculus based on a topological duality theorem in nominal sets. A novel interpretation of lambda is given in terms of adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is necessary).
A definable type of a first-order theory is the same as a section (retraction) of the simplicial path space (decalage) of its space of types viewed as a simplicial topological space; as is well-known, in the category of simplicial sets such sections correspond to homotopies contracting each connected component. Without the simplicial language this is stated in Exercise 8.3.3 in the model theory textbook [Tent-Ziegler], which defines a bijection between the set of all $1$-types definable over a parameter set $B$ and the set of all "coherent" families of continuous sections $π_n:S^T_n(B)\to S^T_…
We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no perfect set P such that |(eta+B) cap (nu+B)|>5 for any distinct eta,nu from P. This answers two questions from our previous works.
In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,ρ)$ iff $|\bigcap \mathcal E'|<ρ$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set $Y\subset \bigcup\mathcal E$ is a "vertex cover" of $\mathcal E$ iff $E\cap Y\ne \emptyset$ for each $E\in \mathcal E$. A vertex cover $Y$ is "minimal" iff no proper subset of $Y$ is vertex cover. If $A$ is a set and $S$ is a set of cardinals, write $$ {[A]}^{S}=\{B\subset A: |B|\in S\}. $$ If $λ$ and $ρ$ are cardinals, $S$ is a set of cardinals, $k\in ω$, then we write $$\mathb…
All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\bullet$ for every infinite cardinal $κ$ there is a space of size $κ$ in which fewer than $cf(κ)$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $κ$ iff $ω\le κ\le 2^{\mathfrak c}$. A space with at least two non-isolated points is called "strongly anti-Urysohn" $($SAU in short$)$ iff any two infinite closed sets in it intersect. We prove that $\bullet$ if $X…
Given a target set $A\subseteq \mathbb{R}^d$ and a real number $β\in (0,1)$, McMullen introduced the notion of $A$ being an absolutely $β$-winning set. This involves a two player game which we call the $β$-McMullen game. We consider the version of this game in which the parameter $β$ is removed, which we call the no-$β$ McMullen game. More generally, we consider the game with respect to arbitrary norms on $\mathbb{R}^d$, and even more generally with respect to general convex sets. We show that for strictly convex sets in $\mathbb{R}^d$, polytopes in $\mathbb{R}^d$, and general convex sets in $…
We discuss two main ways in comparing and evaluating the size of sets: the "Cantorian" way, grounded on the so called Hume principle (two sets have equal size if they are equipotent), and the "Euclidean" way, maintaining Euclid's principle "the whole is greater than the part". The former being deeply investigated since the very birth of set theory, we concentrate here on the "Euclidean" notion of size (numerosity), that maintains the Cantorain defiitions of order, addition and multiplication, while preserving the natural idea that a set is (strictly) larger than its proper subsets. These numer…
The main purpose of this paper is to study \emph{$e$-separable spaces}, originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$ $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. We primarily focus on the behaviour of $e$-separable spaces under products and the cardinal invariants that are naturally related to $e$-separable spaces. Our main results show that the statement "there is a product of at most $\mathfrak c$ many $e$-separable spaces that fails to be $e$-separable'" is equiconsistent with the existence of a weakly compact card…
In this note we show that a set is provably $Δ^0_2$ in the fragment $IΣ_n$ of arithmetic iff it is $IΣ_n$-provably in the class $D_α$ of $α$-r.e. sets in the Ershov hierarchy for an $α<_{ε_0} ω_{1+n}$, where $<_{ε_0}$ denotes a standard $ε_0$-ordering. In the Appendix it is shown that a limit existence rule $(LimR)$ due to Beklemishev and Visser becomes stronger when the number of nested applications of the inference rule grows.
One says that a property $P$ of sets of natural numbers can be made into itself iff there is a numbering $α_0,α_1,\ldots$ of all left-r.e. sets such that the index set $\{e: α_e$ satisfies $P\}$ has the property $P$ as well. For example, the property of being Martin-Löf random can be made into itself. Herein we characterize those singleton properties which can be made into themselves. A second direction of the present work is the investigation of the structure of left-r.e. sets under inclusion modulo a finite set. In contrast to the corresponding structure for r.e. sets, which has only maximal…
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
