Fall 2018
The Final Exam is cumulative and covers all material up to and including Nov. 20 (Ax’s Theorem), with the same exception as for Midterm 2.
Here is the final exam I gave 4 years ago. Note that it included a question about first-order arithmetic, because there was more time between the last day of class and the exam than there is this year. I will not post the solutions; if you have questions, please come and talk to me.
Good luck on your other final exams, and Happy Holidays!
Homework
Note: Only a few selected solutions are posted. Please feel free to ask me directly about marked homework.
| Date Due | Assignment | Remarks |
| September 18 | Exercises 2.28, 2.31, 2.45, 2.70 | due at beginning of class |
| October 2 | Exercises 3.9(b), 3.14(a), 3.27 | due at beginning of class |
| October 16 | Exercises 4.7, 4.24, 4.38, 4.54 | due at beginning of class |
| October 30 | Exercises 4.69, 4.76, 4.93, 4.111 | due at beginning of class |
| November 13 | Exercises 5.26, 5.33, 5.37 | due at beginning of class |
| November 30 changed due date! | Homework 6 Solutions | due at beginning of class |
| Problems on definability | Problems on definability |
Covered so far
| Week | Tuesday | Thursday | Friday | Required reading and recommended exercises |
| Sep 4–7 | Construction of propositional formulas | Interpretation of propositional formulas | Logical equivalence | Sections 2.1 to 2.5 and embedded exercises Exercises 2.29, 2.30, 2.35 |
| Sep 10–14 | Logical consequence | Formal proof | Formal proof | Sections 2.6 and 3.2 and embedded exercises Exercises 2.69, 2.75, 2.82, 3.7 |
| Sep 17-21 | Soundness and consistency | Completeness | Compactness | Section 3.3 and embedded exercises Exercises 3.20, 3.22, 3.23 |
| Sep 24-28 | First order formulas | Structures and interpretation of terms | Interpretation of formulas | Sections 4.1 and 4.2 and embedded exercises 4.5, 4.8, 4.9, 4.20, 4.23 |
| Oct 1-5 | Midterm 1 Covers propositional logic Sections 2.1 — 3.3 of the book Solutions | Validity | Free substitution | Section 4.3 and embedded exercises 4.38, 4.40, 4.43, 4.44 |
| Oct 15-19 | Theories | Quantifier elimination | Algebraically closed and real closed fields | Section 4.4 and embedded exercises algebraically closed and real closed fields |
| Oct 22-26 | Substructures | Substructures | Isomorphisms | Sections 4.5 and embedded exercises Exercises 4.103, and 4.109 |
| Oct 29-Nov 2 | Formal predicate logic | Some derivations | Soundness and consistency | Sections 5.2-5.4 and embedded exercises Exercises 5.10, 5.13, 5.15, 5.24 and 5.25 |
| Nov 5-9 | Midterm 2 Covers predicate calculus, up to Oct. 26 From Section 4.4, only the axiom systems discussed in class are included; axioms will be given on the test if needed. Solutions | Completeness, part I | Completeness, part II | Section 5.5 and embeded exercises Exercises 5.32 and 5.35 |
| Nov 12-16 | Downward Löwenheim-Skolem | Compactness | Completeness of algebraically closed fields | Sections 6.1 and 6.4 |
| Nov 19-23 | Ax’s Theorem | class cancelled | First-order arithmetic | |
| Nov 26-30 | Expressibility | Recursive functions | Examples of recursive functions | Proof of Lemma 1 |
| Dec 3-5 | Gödel numbering Gödel’s incompleteness theorem (slides) | Gödel’s incompleteness theorem (notes with proofs) |
General information
| Instructor: | Patrick Speissegger |
| Office: | HH 409A |
| Telephone: | extension 23430 |
| E-mail: | speisseg@math.mcmaster.ca |
| Office Hours: | Tuesday and Thursday 10:00–11:00, or by appointment |
| Lectures: | Tuesday, Thursday and Friday 11:30–12:20 in HH 312 (the room has been changed as of Aug. 30). |
| Text: | Propositional and predicate calculus, a model of argument, by Derek Goldrei, Springer-Verlag London 2005, ISBN-13 978-1-85233-921-0. This book is available for download through the McMaster Library website (access from campus or via login) |
| Assessment: | Your grade will be based on six homework assignments, in-class participation, one or two in-class midterms and the final exam (check back for details). The distribution is as follows: homework 20%, in-class participation 20%, midterm(s) 20% and final 40%. The instructor reserves the right to change the weight of any portion of this marking scheme. |
| Homework: | Assigned every other week; collected on the due date in class, at the beginning of class. |
| In-class participation: | There are a lot of reading assignments in this class. Discussion of the read material is an integral part of class, especially the exercises embedded in the text. |
| Midterms: | Tuesday, Oct. 2 and Tuesday, Nov. 6 in class (tentative) |
| Final: | to be announced on Mosaic |
| Missed academic work: | In the event of an absence for medical or other reasons, students should review and follow the Academic Regulation in the Undergraduate Calendar Requests for Relief for Missed Academic Term Work. |