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**1 - 4**of**4**### Non-Standard Models of Arithmetic: a Philosophical and Historical perspective

, 2010

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### Recommended Text: Introduction to Model Theory

"... After a couple of weeks to introduce the fundamental concepts and set the context (material chosen from the first three chapters of the text), the course will proceed with the development ..."

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After a couple of weeks to introduce the fundamental concepts and set the context (material chosen from the first three chapters of the text), the course will proceed with the development

### Arithmetic and the Incompleteness Theorems

, 2000

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### TITLE: A History ofthe Theory of Types with Special Reference to Developments After the Second Edition of Principia Mathematica

"... This thesis traces the development ofthe theory of types from its origins in the early twentieth century through its various forms until the mid 1950's. Special attention is paid to the reception of this theory after the publication of the second edition of Whitehead and Russell's Principi ..."

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This thesis traces the development ofthe theory of types from its origins in the early twentieth century through its various forms until the mid 1950's. Special attention is paid to the reception of this theory after the publication of the second edition of Whitehead and Russell's Principia Mathematica. We examine how the theory of types declined in influence over four decades. From being in the 1920s the dominant form of mathematical logic, by 1956 this theory had been abandoned as a foundation for mathematics. The use and modification ofthe theory by logicians such as Ramsey, Carnap, ChurctU Quine, Gddel, and Tarski is given particular attention. Finally, the view of the theory of types as a many-sorted first-order theory in the 1950's is discussed. It was the simple theory, as opposed to the ramified theory of types that was used almost exclusively during the years following the second edition of Principia. However, it is shown in this thesis that in the 1950's a revival of the ramified theory of types occurred. This revival of ramified-type theories coincided with the consideration of cumulative type hierarchies. This is most evident in the work of Hao Wang and John Myhill. The consideration of cumulative tlpe-hierarchies altered the form of the theory of types in a substantial way. The theory was altered even more drastically by being changed from a many-sorted theory into a one-sorted theory. This final "standardization" of the theory of types in the mid 1950's made it not much different from first-order Zermelo-Fraenkel set-theory. The theory oftypes, whose developments are traced in this thesis, therefore lost its prominence as the foundation for mathematics and logic. ur