Studies in Pure Mathematics

Ryan Flannery
Computer Science PhD Student and Math Nerd
ryan.flannery@gmail.com
http://www.ryanflannery.org 		Neal Hogan
Philosophy PhD Student & Logic/Math Nerd
hogannp@email.uc.edu
http://www.nealhogan.net
Professor John Schlipf
Professor/Advisor & Fearless Leader
john.schlipf@uc.edu
http://www.ececs.uc.edu/~schlipf

This page is a work in progress. Check back regularly for update and new notes.

After completing the Mathematical Logic course sequence at the University of Cincinnati, which include three courses (the first two focussing on basic mathematical logic, culminating with Gödel's incompleteness theorem, and the third course focussing on set theory), Neal Hogan and myself decided to form an informal study group to continue our studies. Our studies focus in various fields that, collectively, are commonly called the foundations of mathematics (including mathematical logic, model theory, set theory, domain theory, category theory, lambda calculus, and more). This page is a summary of our studies including all of the notes we have put together thus far and a list of internet resources that we have found usefull. Feel free to download and use/modify any of the papers you see below. If you find any errors, please let either of us know.

Useful Internet Resources

Introductory References Historical References General References Online

Group Documentation and Downloads

Mathematical Logic & Model Theory

  • (tex) (dvi) (ps) (pdf) Soundness and Completeness for Propositional Logic notes (by John Schlipf)
  • (tex) (dvi) (ps) (pdf) Sigma-1 & Pi-1 Definability, Recursiveness, and Incompleteness (by John Schlipf)
  • (tex) (dvi) (ps) (pdf) Work in Progress Herbert Enderton's A Mathematical Introduction to Logic, selected topics from Chapters 2 and 4 (by Ryan Flannery)

Zermelo-Fraenkel Set Theory

  • (tex) (dvi) (ps) (pdf) Keith Devlin's The Joy of Sets Chapter 2, the Zermelo-Fraenkel Axioms (by Ryan Flannery)
  • (tex) (dvi) (ps) (pdf) Keith Devlin's The Joy of Sets Chapter 3, Ordinal and Cardinal Arithmetic (by Neal Hogan)
  • (tex) (dvi) (ps) (pdf) Keith Devlin's The Joy of Sets Chapter 5, the Axiom of Constructibility (by Ryan Flannery)
  • (tex) (dvi) (ps) (pdf) This paper outlines Gödel's half of the proof that GCH is undecidable in ZF Set Theory (by Ryan Flannery)

Lambda Calculus

  • (tex) (dvi) (ps) (pdf) Alonzo Church's The Calculi of Lambda Conversion (by Ryan Flannery)
  • (tex) (dvi) (ps) (pdf) H.P. Barendregt's The Lambda Calculus Chapters 1 through 5 (by Neal Hogan)

Category Theory

  • (tex) (dvi) (ps) (pdf) Work in Progress Benjamin C. Pierce's Basic Category Theory for Computer Scientists (by Ryan Flannery)

Miscellaneous Topics

  • (tex) (dvi) (ps) (pdf) Dana Scott's A Mathematical Theory of Computation (by Neal Hogan)
  • (tex) (dvi) (ps) (pdf) Work in Progress Alfred Tarski's Concept of Truth in Formalized Languages (by Ryan Flannery)