arXiv subjects

Fulgencio Lopez

Publications and source records attributed to Fulgencio Lopez.

3 recordsLinked to original sources

arXiv2018

Forcing and Construction Schemes

We investigate forcing and independence questions relating to construction schemes. We show that adding $κ\geqω_1$ Cohen reals adds a capturing construction scheme. We study the weaker structure of $n$-capturing construction schemes and show that it is consistent to have $n$-capturing construction schemes but no $(n+1)$-capturing construction schemes. We also study the relation of $n$-capturing with the $m$-Knaster hierarchy and show that MA$_{ω_1}($K$_m)$ and $n$-capturing are independent if $n\leq m$ and incompatible if $n>m$.

math.LO↗

arXiv2016

Banach spaces from a construction scheme

We construct a Banach space $\mathcal X_\varepsilon$ with an uncountable $\varepsilon$-biorthogonal system but no uncountable $τ$-biorthogonal system for $τ<\varepsilon$. In particular the space have no uncountable biorthogonal system. We also construct a Banach space $\mathcal X_K$ with an uncountable $K$-basic sequence but no uncountable $K'$-basic sequence, for $1\leq K'<K$. A common feature of these examples is that they are both constructed by recursive amalgamations using a single construction scheme.

math.LO↗

arXiv2016

Trees and gaps from a construction scheme

We present natural constructions of trees and gaps using a quite general construction scheme. In particular, we solve a natural problem about $(ω_1, ω_1)$-gaps. As it is well known $(ω_1, ω_1)$-gaps can sometimes be filled in $ω_1$-preserving forcing extensions of the set-theoretic universe. There are two natural conditions, dubbed $S$ and $T$ below, that guarantee the existence of such forcing extensions. The condition $T$ is a natural strengthening of the condition $S$ and was motivated by the numerous analogies between $(ω_1,ω_1)$-gaps and certain trees of height $ω_1.$ It turns out that the condition $S$ is in fact equivalent to the existence of such forcing extensions but we show that the condition $T$ is strictly stronger by proving that it is consistent that there are fillable $(ω_1, ω_1)$-gaps (i.e., S-gaps) but no T-gaps.

math.LO↗