Gregory H. Moore

The Origins of Forcing

Abstract This paper is a history of the origins and early development of forcing. After mentioning independence and consistency questions prior to Godel, the paper discusses constructibility and Godels attempts to prove the independence of the Axiom of Choice and the Continuum Hypothesis. The paper explores how Cohen created forcing and how Feferman, Levy, Scott, and especially Solovay developed it further.

The Origins of Russell’s Paradox: Russell, Couturat, and the Antinomy of Infinite Number

From 1897 to 1913, during the entire period when Russell made his major contributions to mathematical logic, he corresponded regularly with the French philosopher, Louis Couturat. Almost 200 letters passed between them, ranging over a myriad of topics, from the politics of the Boer War to the usefulness of an international language. The heart of this correspondence, however, is not its political themes but its logical, mathematical, and philosophical ones.

Historians and Philosophers of Logic: Are They Compatible? The Bolzano-Weierstrass Theorem as a Case Study

This paper combines personal reminiscences of the philosopher John Corcoran with a discussion of certain conflicts between historians of logic and philosophers of logic. Some mistaken claims about the history of the Bolzano-Weierstrass Theorem are analyzed in detail and corrected.

The Prehistory of Infinitary Logic: 1885–1955

Traditionally, logic was restricted to proofs having a finite number of steps and to expressions of finite length. Around 1954–56, infinitely long formulas entered the mainstream of mathematical logic through the work of Henkin, Karp, Scott, and Tarski. Soon Hanf and Tarski used such logics to settle negatively the 30-year-old problem of whether the first strongly inaccessible cardinal is measurable, a result Tarski communicated to the first LMPS congress in 1960. Infinitary logic continues to be fertile in unexpected ways, as shown by Kolaitis at the present congress.

Proof and the infinite

ConclusionWhen modern infinitary logic arose in the mid 1950s, it was motivated primarily by the desire to extend first-order logic to a stronger logic that would retain certain desirable properties of first-order logic. Those who invented this infinitary logic showed little awareness of their predecessors work using infinitely long formulas, such as that of Löwenheim and Carnap. But, thanks to this modern infinitary logic, the notion of formal proof was enlarged in a fundamental way.During the 20th century the notion of formal proof has been one of the most fertile and important notions in mathematical logic. The distinction between syntactic and semantic notions (for example, proof vs. truth, consistency vs. satisfiability, theorem vs. logical consequence) is something that everyone well educated in mathematics should be aware of. While educators can reasonably differ as to when students should learn these notions, they can hardly deny the fact that students of mathematics should understand formal proof — its uses and its limitations. Infinitary logic is important in overcoming certain of these limitations, and so has a significant place in the education of all those who wish to understand mathematics in depth.