Souslin Quasi-Orders and Bi-Embeddability of Uncountable Structures.

Format:

Book

Author/Creator:

Andretta, Alessandro.

Contributor:

Ros, Luca Motto.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.277

Language:

English

Subjects (All):

Set theory.

Topological spaces.

Embeddings (Mathematics).

Physical Description:

1 online resource (204 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"We provide analogues of the results from Friedman and Motto Ros (2011) and Camerlo, Marcone, and Motto Ros (2013) (which correspond to the case) for arbitrary -Souslin quasi-orders on any Polish space, for an infinite cardinal smaller than the cardinality of R. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size , the isometric embeddability relation between complete metric spaces of density character , and the linear isometric embeddability relation between (real or complex) Banach spaces of density "-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. What we knew

1.2. What we wanted

1.3. What we did

1.4. How we proved it

1.5. Classification of non-separable structures up to bi-embeddability

1.6. Organization of the paper, or: How (not) to read this paper

1.7. Annotated content

Chapter 2. Preliminaries and notation

2.1. Basic notions

2.2. Choice and determinacy

2.3. Cardinality

2.4. Algebras of sets

2.5. Descriptive set theory

2.6. Trees and reductions

Chapter 3. The generalized Cantor space

3.1. Basic facts

3.2. *More on 2^{ }

Chapter 4. Generalized Borel sets

4.1. Basic facts

4.2. Intermezzo: the projective ordinals

4.3. *More on generalized Borel sets

Chapter 5. Generalized Borel functions

5.1. Basic facts

5.2. *Further results

Chapter 6. The generalized Baire space and Baire category

6.1. The generalized Baire space

6.2. Baire category

Chapter 7. Standard Borel -spaces, -analytic quasi-orders, and spaces of codes

7.1. -analytic sets

7.2. Spaces of type and spaces of codes

Chapter 8. Infinitary logics and models

8.1. Infinitary logics

8.2. Some generalizations of the Lopez-Escobar theorem

Chapter 9. -Souslin sets

9.1. Basic facts

9.2. More on Souslin sets and Souslin cardinals

9.3. Souslin sets and cardinals in models with choice

9.4. Souslin sets and cardinals in models of determinacy

Chapter 10. The main construction

10.1. The combinatorial trees ₀ and ₁

10.2. The combinatorial trees _{ }

Chapter 11. Completeness

11.1. Faithful representations of -Souslin quasi-orders

11.2. The quasi-order ≤_{max} and the reduction Σ_{ }

11.3. Reducing ≤_{max}^{ } to \embeds^{ }_{\CT}

11.4. Some absoluteness results

Chapter 12. Invariant universality.

12.1. An \LL_{ ⁺ }-sentence \Uppsi describing the structures _{ }.

12.2. A classification of the structures in \Mod^{ }_{\Uppsi} up to isomorphism

12.3. The invariant universality of \embeds^{ }_{\CT}

12.4. More absoluteness results

Chapter 13. An alternative approach

13.1. Completeness

13.2. Invariant universality

Chapter 14. Definable cardinality and reducibility

14.1. Topological complexity

14.2. Absolutely definable reducibilities

14.3. Reducibilities in an inner model

Chapter 15. Some applications

15.1. \bSigma¹₂ quasi-orders

15.2. Projective quasi-orders

15.3. More complex quasi-orders in models of determinacy

15.4. \Ll(\R)-reducibility

Chapter 16. Further completeness results

16.1. Representing arbitrary partial orders as embeddability relations

16.2. Other model theoretic examples

16.3. Isometry and isometric embeddability between complete metric spaces of density character

16.4. Linear isometry and linear isometric embeddability between Banach spaces of density

16.5. *Further results on the classification of nonseparable metric and Banach spaces

Indexes

Index

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781470470975

1470470977

OCLC:

1343248959