S ebastiano T hei . Singular cardinals through the lens of Shelah’s pcf theory and Prikry-type forcings . University of Udine. 2025. Supervised by Vincenzo Dimonte. MSC 03E35, 03E04, 03E55, 03E15. Keywords: large cardinals, pcf theory, forcing, HOD hypothesis, scale.
Year: 2025
Type: article
Abstract: The development of the forcing method has shown that several key questions regarding infinite sets cannot be settled under ZFC alone. The most widely supported view is that this undecidability simply reflects the limitations of ZFC in addressing all mathematical problems. This perspective has motivated an extensive search for new axioms—the so-called large cardinal axioms —which, when added to ZFC, yield a deeper and more robust understanding of the set-theoretic universe. Along this line, the dissertation is divided into three thematic blocks, each of them framed within these extensions of ZFC: • Very large cardinals at the threshold of Kunen inconsistency, with a focus on elementarity and cardinal correctness (Chapter 2). • Generalized Descriptive Set Theory at singular strong limit cardinals of countable cofinality, with a focus on two regularity properties (Chapter 3). • Covering lemmas and Woodin’s HOD Dichotomy through the lens of Shelah’s pcf theory (Chapter 4). Specifically, Chapter 2 establishes an inconsistency result using tools from singular cardinal combinatorics and Shelah’s pcf theory, proving the nonexistence of cardinal preserving elementary embeddings into V and establishing thereby a limitation in the hierarchy of large cardinal axioms. The proof is based on the notion of good scales and its connection with Jónsson cardinals. Chapter 3 proves a consistency result obtained via a Prikry-type forcing construction, providing a singular-cardinal analogue of Solovay’s theorem. This thematic block is inspired by Woodin’s Axiom I 0 $I_0$ upper I 0 which provides the appropriate axiomatic framework to develop Generalized Descriptive Set Theory in generalized Baire/Cantor spaces at singular cardinals. Concretely, we construct a model of ZFC where κ $\kappa $ kappa is a strong limit singular cardinal with countable cofinality, and every subset of ω κ ${}^\omega \kappa $ Superscript omega Baseline kappa in L ( V κ + 1 ) $L(V_{\kappa +1})$ upper L left parenthesis upper V Subscript kappa plus 1 Baseline right parenthesis has both the κ $\kappa $ kappa -Perfect Set Property and the U → $\vec {\mathcal {U}}$
Source: Bulletin of Symbolic Logic
Author: Sebastiano Thei
Institution: University of Udine
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W7116802182
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