Course description:

This course covers logical metatheory and elementary meta-mathematics. Topics include completeness and consistency proofs for first-order logic, formal semantics, elementary number theory (especially Robinson and Peano arithmetics), and Gödel’s incompleteness theorems and related results.

Prerequisites:

Phil 310 (Intermediate Logic) or equivalent and solid grasp of high school algebra, or consent of instructor. You must also be prepared for a lot of challenging work. Fair warning: most students find this course much more difficult than their earlier logic courses.

Contact information:

Prof. Klement’s email is klement@umass.edu, and his office is South College E319. Office hours are Mondays 10am–11am, Thursdays 11:30am–12:30pm, or click here to schedule an appointment. or schedule an appointment online at https://logic.umasscreate.net/appts/?view=klement

Moodle page:

We have a site on the UMass Moodle LMS (https://umass.moonami.com/), where you can view your grades and download course materials.

Lecture notes:

I have prepared a lengthy document containing all my lecture notes for the course. It is available both through Moodle and at the link just given. on our public-facing webpage for the course: https://people.umass.edu/klement/513/. You are expected to have them available (print them beforehand or access them on a device) during class. I will not distribute these to you, and lectures will be impossible to follow without them. Page numbers to be covered are on the schedule, but we may get ahead of schedule, so print ahead of time.

Textbook:

We will be using the Open Logic Text, a freely available, “copyleft-ed”, open educational resource, which anyone can modify or contribute to. It is available to download from Moodle or from the Open Logic Project website. the Open Logic Project website: http://builds.openlogicproject.org/

The section numbers in the schedule below come from the “Clean Version”, Revision ec6994c, dated 2022-08-13, but new versions are posted often.

Requirements and grading:

Your final grade is determined by (1) four homework bundles (22% each / 88% total); (2) class attendance and participation (12%).

There are homework exercises assigned every class: a total of 36. The actual assignments are found scattered in the lecture notes, alongside the material they involve. Assigned exercises will be collected in four “bundles”:
  • Bundle 1 (HW 1–8) is due Thurs. September 29th
  • Bundle 2 (HW 9–17) is due Thurs. October 20th
  • Bundle 3 (HW 18–28) is due Thurs. November 17th
  • Bundle 4 (HW 29–36) is due Tues. December 20th

You are also expected to attend class regularly, prepared to discuss the material, and participate by asking questions, making comments and responding to the questions and comments of others.

Policies:

Homework may be handwritten. You may collaborate with your peers on homework assignments provided you write up your final answers entirely on your own and do not do any copying.

Schedule

Warning! Everything about this schedule can and will change.

Date Material (lecture notes pages) Book sections Homework
Tu 6 Sept Course introduction
Th 8 Sept Metatheory, set theory (pp. 1–3) §§1.1–4.9 HW1
Tu 13 Sept Mathematical induction, propositional logic (pp. 4–6) §§71.1–71.5; §§7.1–7.5 HW2, HW3
Th 15 Sept Syntax/semantics of first-order logic (pp. 7–11) §§15.1–15.8; §§16.1–16.3 HW4, HW5
Tu 20 Sept More on semantics of first-order logic (pp. 11–13) §§16.4–16.7 HW6, HW7
Th 22 Sept Tableaux for first-order logic (pp. 13–16) §§21.1–21.6 HW8
Tu 27 Sept Axiomatic deductions (pp. 16–20) §§22.1–22.6 HW9, HW10
Th 29 Sept Deduction theorem and corollaries (pp. 20–23)
HW 1–8 due		 §§22.7–22.11 HW11
Tu 4 Oct Soundness and consistency (pp. 23–27) §22.12; §§34.1–34.2 HW12
Th 6 Oct Lemmas for completeness (pp. 27–30) §§23.1–23.6 HW13
Tu 11 Oct Completeness, compactness (pp. 30–32) §§23.8–23.11 HW14
Th 13 Oct Identity logic (pp. 32–35) §22.13; §23.7 HW15, HW16
Tu 18 Oct Unit 1 review and questions HW17
Th 20 Oct Theories; Peano and Robinson arithmetics (pp. 36–39)
HW 9–17 due.		 §§17.1–17.3; §§33.1–33.4 HW18, HW19
Tu 25 Oct Naïve foundations (pp. 40–41) §1.6 HW20
Th 27 Oct Numerals; Q vs. PA (pp. 41–46) §35.1; §35.5 HW21
Tu 1 Nov Recursive functions (pp. 46–50) §§29.1–29.7 HW22, HW23
Th 3 Nov More on recursive functions (pp. 50–53) §§29.8–29.13 HW24, HW25
Tu 8 Nov Representing functions in Q (pp. 53–57) §35.6–35.9 HW26, HW27
Th 10 Nov Representing recursion (pp. 57–60) §§35.3–35.4 HW28
Tu 15 Nov Arithmetization of syntax (pp. 60–65) §§34.3–34.5; §34.8; §35.2 HW29, HW30
Th 17 Nov Fixed point theorem; Gödel’s first theorem (pp. 65–70)
HW 18–28 due.		 §§37.1–37.3 HW31, HW32
Tu 22 Nov No class. Friday class schedule.
Th 24 Nov No class. Thanksgiving break.
Tu 29 Nov Gödel-Rosser and Gödel’s second theorem (pp. 71–73) §§37.4–37.7 HW33
Th 1 Dec Löb’s theorem; Tarski’s theorem (pp. 73–75) §§37.8–37.9 HW34
Tu 6 Dec Recursive undecidability (pp. 75–78) §35.10; §36.4; §36.9 HW35
Th 8 Dec Unit 2 review and questions HW36
Tu 20 Dec End of finals week.
HW 29–36 due.

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