On construction schemes: building the uncountable from finite pieces

TL;DR

This thesis develops a structural analysis of construction schemes to build uncountable combinatorial objects, demonstrating their independence from standard set theory axioms.

Contribution

It introduces new methods for constructing uncountable structures using finite pieces and analyzes their axioms, expanding understanding of combinatorial set theory.

Findings

Constructed several uncountable structures independent of ZFC

Developed a framework for analyzing construction schemes

Connected combinatorial objects with set-theoretic independence

Abstract

In this Phd. thesis, a structural analysis of construction schemes is developed. The importance of this study will be justified by constructing several distinct combinatorial objects which have been of great interest in mathematics. We then continue the study of capturing axioms associated to construction schemes. From them, we construct several uncountable structures whose existence is known to be independent from the usual axioms of Set Theory.