Rosser's Theorem via Turing Machines (2011)
The post delves into Rosser's Theorem, a potent extension of Gödel's Incompleteness Theorems, showcasing self-reference in formal systems. It elucidates Rosser's stronger disproof concept and its impact on system consistency. Additionally, it links Rosser's Theorem to the halting problem's unsolvability, underlining computational insights and Gödel's lasting influence on computability theory.
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The blog post discusses Rosser's Theorem in the context of Turing machines and Gödel's Incompleteness Theorems. It explains how Gödel's First Incompleteness Theorem leads to self-referential paradoxes in formal systems. The post introduces Rosser's Theorem as a stronger version of Gödel's Theorem, highlighting the concept of a sentence with a shorter disproof for every proof. It explores the implications of Rosser's Theorem on the consistency and completeness of formal systems. The author presents a Turing machine-based proof of Rosser's Theorem, connecting it to the unsolvability of the halting problem. The post emphasizes the significance of computational perspectives in understanding these theorems and concludes with reflections on the enduring relevance of Gödel's Theorem in computability theory.
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5 commentsIt's beautiful, elegant, and easy to understand. I was introduced to the proof by a note in Sipser's text.
In number theory, Rosser's theorem states that the {\displaystyle n}th prime number is greater than log {\displaystyle n\log n}, where log{\displaystyle \log } is the natural logarithm function. It was published by J. Barkley Rosser in 1939.
Wiki refers to what Scott Aaronson is calling Rosser's theorem as "Rosser's trick"
Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation".
This textbook is an astonishing piece of work of an astonishing mathematical genius. The amount of all kinds of weird (and useful? I'm not much of a logician myself) logical stuff in it is just mind-boggling, just as its writing style. You can keep come back to this book for years and find something new in it every time.
Barendregt's book on lambda calculus is another example of such a book.
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