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Zermelo's Axiom of Choice: Its Origins, Development, and Influence
Prologue.- 1 The Prehistory of the Axiom of Choice.- 1.1 Introduction.- 1.2 The Origins of the Assumption.- 1.3 The Boundary between the Finite and the Infinite.- 1.4 Cantor's Legacy of Implicit
The emergence of first-order logic
To most mathematical logicians working in the 1980s, first-order logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician
Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory
What has been the historical relationship between set theory and logic? On the one hand, Zermelo and other mathematicians developed set theory as a Hilbert-style axiomatic system. On the other hand,
The axiomatization of linear algebra: 1875-1940
Modern linear algebra is based on vector spaces, or more generally, on modules. The abstract notion of vector space was first isolated by Peano (1888) in geometry. It was not influential then, nor
Hilbert and the emergence of modern mathematical logic
Hilbert's unpublished 1917 lectures on logic, analyzed here, are the beginning of modern metalogic. In them he proved the consistency and Post-completeness (maximal consistency) of propositional
The origins of Zermelo's axiomatization of set theory
This paper argues that Zermelo was primarily motivated, not by the paradoxes, but by the controversy surrounding his 1904 proof that every set can be well-ordered, and especially by a desire to preserve his Axiom of Choice from its numerous critics.
The Dual Cantor-Bernstein Theorem and the Partition Principle
It is shown that the Refined Dual Cantor-Bernstein Theorem is equivalent to the Axiom of Choice.