Construction schemes: transferring structures from $ω$ to $ω_1$

Abstract: A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of capturing axioms associated to construction schemes. From them, we deduce the existence of several uncountable structures which are known to be independent from the usual axioms of Set Theory. Lastly, we prove that the capturing axiom $FCA(part)$ is implied by Jensen's $\Diamond$ principle.