Systems of Logic Based on Ordinals

Systems of Logic Based on Ordinals was Alan Turing's PhD thesis at Princeton University, published in 1938. The work examines fundamental questions in mathematical logic and introduces the concept of ordinal logic. Turing builds on Gödel's incompleteness theorems to explore how adding new axioms to formal systems affects their power and limitations. The text presents detailed proofs and frameworks for analyzing logical systems through the lens of ordinal numbers. The book establishes connections between computability theory, proof theory, and transfinite ordinal numbers. Turing develops the notion of oracle machines and relative computability as tools for comparing the strength of different logical systems. At its core, this work represents an attempt to understand the boundaries between provable and unprovable mathematical statements, and the nature of mathematical truth itself. The concepts introduced continue to influence modern research in mathematical logic and theoretical computer science.

There appear to be very few public reader reviews of this book, as it was Turing's PhD thesis rather than a widely published work. Most discussion comes from academic papers citing or analyzing it. Readers highlight: - Clear explanation of oracle machines and relative computability - Mathematical rigor in examining logical systems - Historical significance in computability theory Common criticisms: - Dense technical writing makes it inaccessible to non-specialists - Limited availability of the original text - Lack of modern notation/formatting in reproductions No ratings found on Goodreads, Amazon, or other consumer review sites. The work is primarily referenced and reviewed in academic journals and mathematics publications. Professor Martin Davis noted it was "years ahead of its time" in pioneering relative computability concepts. Solomon Feferman called it "remarkable" but "largely ignored" in early years after publication.

Principia Mathematica by Alfred North Whitehead, Bertrand Russell. A foundational text that establishes mathematical logic from first principles and explores the relationship between mathematics and formal logical systems.

Gödel's Proof by Ernest Nagel, James Newman. A detailed examination of Gödel's incompleteness theorems and their implications for mathematical logic and computability theory.

An Introduction to Mathematical Logic by Elliott Mendelson. This text presents the fundamental concepts of mathematical logic, including formal systems, proof theory, and recursive functions.

Set Theory and Logic by Robert R. Stoll. A systematic development of set theory and its connection to mathematical logic, including ordinal numbers and transfinite arithmetic.

Recursion Theory by Joseph R. Shoenfield. An exploration of computability theory, recursive functions, and their connection to mathematical logic and ordinal hierarchies.

🔹 This 1938 work was Turing's Ph.D. thesis at Princeton University, completed under the supervision of Alonzo Church, and later published in 1939. 🔹 The book introduced the concept of "oracle machines" - theoretical devices that could solve problems beyond the capabilities of standard Turing machines, fundamentally advancing our understanding of computability. 🔹 Turing wrote this groundbreaking work shortly after his more famous 1936 paper "On Computable Numbers," showing his rapid evolution of thought about mathematical logic and computation. 🔹 The thesis explores the concept of "ordinal logics" which Turing developed as a potential way to overcome Gödel's incompleteness theorems - a bold attempt to extend the boundaries of mathematical proof. 🔹 The ideas presented in this work laid crucial groundwork for later developments in artificial intelligence and machine learning, particularly in understanding hierarchical levels of problem-solving capability.