Side Conditions of Models of Two Types and High Forcing Axioms

Rodriguez, Curial Gallart (2024) Side Conditions of Models of Two Types and High Forcing Axioms. Doctoral thesis, University of East Anglia.

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Abstract

The present dissertation is a contribution to the areas of combinatorial set theory and high forcing axioms through the technique of forcing with side conditions. We introduce new forcing notions consisting of symmetric systems of models of two types, which can be seen as generalizations of both Neeman's chains of elementary submodels of two types and Aspero and Mota's symmetric systems of countable elementary submodels. After a preliminary chapter in which we establish the notation and cover the background material required for what follows, we develop the theory of the pure side condition forcings and prove their main properties. The first application of this technique is in the area of combinatorial set theory. We partially answer a question of Hajnal and Szentmiklossy from the 1990s, by forcing a strong chain of subsets of w1 of length w3, improving earlier results of Koszmider and Velickovic-Venturi. In the final chapter we introduce finite support forcing iterations with symmetric systems of models of two types as side conditions in the sense of Aspero and Mota. We isolate a class of forcing notions naturally associated with these iterations and prove the consistency of its forcing axiom, which is compatible with 2N0 > N2. This class of posets, which is a subclass of Neeman's high analog of the class of proper forcings, can be seen as a generalization of Aspero and Mota's classes of finitely proper forcings and forcings with the N1:5-chain condition.

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Item Type:	Thesis (Doctoral)
Faculty \ School:	Faculty of Science > School of Mathematics (former - to 2024)
Depositing User:	Kitty Laine
Date Deposited:	12 Nov 2024 12:05
Last Modified:	12 Nov 2024 12:05
URI:	https://ueaeprints.uea.ac.uk/id/eprint/97643
DOI:

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