Linear Orders and the Real Line

TL;DR

This paper explores various types of linear orders, including dense, separable, and complete orders, and discusses key theorems like Cantor's and Suslin's, highlighting their properties, relationships, and historical context.

Contribution

It provides a comprehensive overview of linear order types, characterizations, and the relationships between Suslin lines, trees, and other order structures, including proofs and historical insights.

Findings

Cantor's Theorem characterizes (Q,<) uniquely among countable dense linear orders.

(R,<) is the only separable complete dense linear order without endpoints.

Suslin lines and trees are shown to be equivalent concepts.

Abstract

Definitions of dense linear orders (with/without endpoints), separable linear orders, complete linear orders, the countable chain condition for linear orders, a Suslin line/Suslin tree and Suslin's problem Statement and proof of Cantor's Theorem characterizing (Q,<) as the only countable dense linear order without endpoints, up to isomorphism, the corollary which characterizes (R,<) as the only separable complete dense linear order without endpoints, every countable linear order embeds into (Q,<) (and thus, into (R,<)). Explanation of why Suslin lines and Suslin trees are equivalent, what an Aronszjan line/tree is, how it's a weakening of a Suslin line/tree. History and independence of Suslin's problem.