Trees and gaps from a construction scheme

TL;DR

This paper introduces a general construction scheme for trees and gaps, solves a problem about $(, )$-gaps, and explores conditions that guarantee the existence of certain forcing extensions, revealing differences between conditions $S$ and $T$.

Contribution

It provides a natural construction scheme for trees and gaps, clarifies the relationship between conditions $S$ and $T$, and demonstrates the existence of fillable gaps without T-gaps.

Findings

Condition $S$ is equivalent to the existence of certain forcing extensions.

Condition $T$ is strictly stronger than $S$.

There are fillable $(, )$-gaps without T-gaps.

Abstract

We present natural constructions of trees and gaps using a quite general construction scheme. In particular, we solve a natural problem about $(\omega_1, \omega_1)$-gaps. As it is well known $(\omega_1, \omega_1)$-gaps can sometimes be filled in $\omega_1$-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 $(\omega_1,\omega_1)$-gaps and certain trees of height $\omega_1.$ 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 $(\omega_1, \omega_1)$-gaps (i.e., S-gaps) but no T-gaps.