13-1 Pure Mathematics Set Theory, Logic, Recursion, Computability, Model Theory

Theoretical Physics with Mark Weitzman

18 Mar 202315:11

EducationalLearning

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TLDRIn this video, Mark Weissman revisits his popular list of pure mathematics books, focusing on foundational topics such as set theory, logic, recursion, computability, and model theory. He recommends undergraduate books by Herbert Enderton for set theory and logic, and 'Intermediate Set Theory' for deeper exploration. Weissman also highlights 'The Joy of Sets' by Keith Devlin and 'A Tour Through Mathematical Logic' for contemporary insights. Graduate-level recommendations include 'Set Theory and the Continuum Hypothesis' by Paul Cohen and 'Set Theory: An Introduction to Independence Proofs' by Kenneth Kunen. Weissman emphasizes the importance of understanding foundational mathematics for those interested in the subject's core.

Takeaways

📚 Mark Weissman introduces his video on adding to his list of pure mathematics books, focusing on foundational topics such as set theory, logic, recursion, computability, model theory, and number theory.
🎓 He recommends starting with undergraduate books on set theory and logic, highlighting Herbert Enderton's works as excellent resources for beginners.
🔍 Weissman mentions 'Intermediate Set Theory' as a book that covers advanced topics like forcing and constructable sets, but notes that it becomes challenging towards the end.
📘 'The Joy of Sets' by Keith Devlin is praised for its coverage of a wide range of topics, including the anti-foundation axiom, which has applications in computer science.
🌐 'A Tour Through Mathematical Logic' is described as a mix of textbook and popular book, providing an overview of modern topics in set theory without going into deep proofs.
📚 Weissman discusses graduate-level books, such as one by Manin, which offers detailed discussions on logic, the Continuum problem, and recursive functions.
📘 He also mentions 'Set Theory and the Continuum Hypothesis' by Paul Cohen, the original work on the subject, which he finds challenging but plans to revisit.
🤖 'Introduction to Boolean Algebras' by Paul Halmos is recommended for those needing a solid foundation in Boolean algebra to understand set theory books.
📘 Kenneth Kunen's book is suggested for those who have already studied set theory and mathematical logic and are looking for a concise review.
🔑 Weissman emphasizes the importance of understanding foundational mathematics for those interested in pure mathematics, providing a list of books to aid in this study.
👨‍🏫 The speaker's passion for foundational mathematics is evident, and he provides a personal touch by sharing his own learning journey and book collection.

Q & A

What is the main topic of the video by Mark Weissman?
-The main topic of the video is to add to the list of pure mathematics books, focusing on the foundations of mathematics, which includes set theory, logic, recursion, computability, model theory, and number theory.
What are the two main interests of Mark Weissman in pure mathematics as mentioned in the video?
-Mark Weissman's two main interests in pure mathematics are foundations of mathematics and number theory.
Which book does Mark Weissman recommend for an undergraduate level understanding of set theory?
-Mark Weissman recommends 'Set Theory' by Herbert Enderton for an undergraduate level understanding of set theory.
What is unique about the book 'A Mathematical Introduction to Logic' by Herbert Enderton according to the video?
-The book 'A Mathematical Introduction to Logic' by Herbert Enderton is unique because it goes through all the details on recursion, representing functions, and proves the incompleteness theorems and undecidability without skipping any details.

Outlines

00:00

📚 Introduction to Pure Mathematics Books

Mark Weissman introduces his video on the topic of pure mathematics books, focusing on his main interests: foundations of mathematics, which include set theory, logic, recursion, computability, model theory, and number theory. He plans to recommend some books he owns, some of which may repeat from a previous video on pure mathematics for physicists. Weissman suggests starting with undergraduate books on set theory and logic, mentioning Herbert Enderton's two excellent books that cover ZFC axioms, the construction of real numbers, cardinals, ordinals, and special topics like König's theorem. He emphasizes the importance of these foundational topics in understanding pure mathematics.

05:02

🔍 Exploring Advanced Set Theory and Logic Books

Weissman continues by discussing more advanced set theory books, including 'Intermediate Set Theory' which covers forcing, constructible sets, and independence results, though he finds the latter part challenging. He also recommends 'The Joy of Sets' by Keith Devlin for its broad coverage and discussions on topics like constructability and independent proofs. Weissman mentions the 'Anti-Foundation Axiom' relevant to computer science and highlights a book that provides an overview of modern mathematical logic, including Goodstein's theorem and Paris-Harrington theorem. He also praises a graduate-level book by Man for its in-depth coverage of logic, the Continuum problem, forcing, and recursive functions, among other topics.

10:02

📘 Graduate-Level Texts and Historical Works on Set Theory

The video script delves into graduate-level texts, featuring a book by Man that Weissman finds comprehensive and fast-paced, covering topics like truth, decidability, the Continuum problem, and recursive functions. Weissman also mentions historical works such as Paul Cohen's original book on the Continuum hypothesis and a book by Raymond Smullyan, known for his expertise in logic. He acknowledges the difficulty of these texts but still considers them essential for a deeper understanding of set theory and logic.

Mindmap

Keywords

💡Pure Mathematics

Pure mathematics refers to the study of mathematical concepts independently of any application outside of mathematics. It is the branch of mathematics that deals with the intrinsic properties of numbers, shapes, and spaces. In the video, the speaker discusses books on pure mathematics, particularly focusing on foundational topics which are central to the theme of the video.

💡Foundations

In the context of the video, 'foundations' refers to the base principles and theories that underpin mathematics. This includes areas such as set theory, logic, recursion, computability, and model theory. The speaker's main interest in pure mathematics lies in these foundational areas, which form the core of the video's discussion.

💡Number Theory

Number theory is a branch of pure mathematics devoted to the study of the integers and their properties. It is one of the speaker's main interests, mentioned alongside foundations. The video suggests that the speaker will add books related to number theory, indicating its relevance to the content.

💡Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. In the video, the speaker recommends books on set theory for those starting to study mathematical foundations, highlighting its importance in understanding the basics of mathematics.

💡Logic

Logic, in mathematics, refers to the study of valid reasoning and the principles of inference. The speaker mentions logic as part of the foundational topics and recommends a specific book that covers the details of mathematical logic at an undergraduate level.

💡ZFC System

The ZFC system, named after Ernst Zermelo and Abraham Fraenkel with the addition of the Axiom of Choice (C), is a standard axiomatization of set theory. The speaker discusses this system as part of the undergraduate set theory book's content, indicating its fundamental role in the study of sets.

💡Incompleteness Theorems

The incompleteness theorems, formulated by Kurt Gödel, are fundamental theorems in the philosophy of mathematics and logic. They establish inherent limitations of every formal axiomatic system. The speaker mentions these theorems as being proved in one of the recommended logic books, emphasizing their importance in the study of mathematical logic.

Highlights

Mark Weissman introduces a video on adding to his book list, focusing on pure mathematics books.

The main interest in pure mathematics discussed is foundations, including set theory, logic, recursion, computability, model theory, and number theory.

Recommendation to start with undergraduate books on set theory and logic, specifically mentioning Herbert Enderton's two excellent books.

Enderton's book on set theory covers ZFC axioms, construction of real numbers, cardinals, ordinals, and special topics including the Erdős-König theorem.

Enderton's mathematical logic book is praised for its detailed coverage and proofs of incompleteness theorems and undecidability.

Introduction of 'Intermediate Set Theory' book that covers basic set theory and advanced topics like forcing and constructable sets.

Recommendation of 'The Joy of Sets' by Keith Devlin for its broad coverage and discussion of the anti-foundation axiom.

Discussion of a book that combines textbook and popular book elements, providing a tour through mathematical logic and modern topics.

A graduate-level book by Manin is highlighted for its detailed approach to logic, the Continuum problem, and recursive functions.

Mention of Paul Cohen's original book on set theory and the Continuum hypothesis, noted for

Transcripts

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