This is a special topics course which introduces students to the key concepts and techniques of:

• First order logic
• Axiomatisation of set theory
• Model theory
• Computability
• Godel's Incompleteness Theorem.

Note: This is an HPC. It emphasises mathematical rigour and proof.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

1. Explain the fundamental concepts from the foundations of mathematics and its role in modern mathematics and applied contexts.
2. Demonstrate accurate and efficient use of logical and set theoretical techniques.
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from the foundations of mathematics.

Examination Material or equipment

If there is an invigilated exam (the plan, COVID permitting), this will allow "all materials" (with the usual provisos about equipment that allows communication).

Required Resources

Full notes (a book really in electronic form) are supplied for the course on Wattle.

The notes can be printed off, but it would be far better to download them onto a laptop or tablet and read them there.

Recommended Resources

Recommended student system requirements 

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

For more information please see https://www.anu.edu.au/students/systems/recommended-student-system-requirements

Staff Feedback

Students will be given feedback in the following forms in this course:

• written comments
• verbal comments
• feedback to whole class, groups, individuals, focus group etc

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Other Information

The course is conducted in a slightly unusual way, more like a reading course than a traditional lecture course. The provided notes are very full and contain both the formal matter of the course and the informal discussion that would usually occur in lectures. There is therefore less contact time allocated than with a normal course, allowing more time to be spent with the notes.

Four contact hours are allocated per week, of which it is only necessary to attend one. These times are used for discussion of the material in the notes and matters and questions arising from them. Participation for at least one hour per week is therefore important.

In 2021, the course will be run entirely remotely, and so the scheduled contact hours will be done via Zoom.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Introduction - A brief introduction to formal languages, formal systems and deductive systems, including an example of a deductive system which does not include logical axioms. The notions of proof, theorem and deduction defined. A brief discussion of metalanguage.
2	Sentential logic - Sentential logic is set up as a purely formal system and developed from that basis. The notion of a semi-formal version of the language is developed. The Deduction Theorem is proved and discussed. The proof that Sentential Logic is decidable is presented using the truth table method. Alternative axiomatisations and equivalence of deductive systems.	Take-home Assignment 1
3	Predicate Logic - Predicate Logic is set up as a purely formal system, First Order Theories are defined and discussed, extending the development in Chapter 2 above, and the standard theorems derived. The Deduction Theorem re-proved in this context. The Choice Rule. The consistency of Pure Predicate Logic. The existence of a complete consistent extension of any consistent first order system.
4	Some first-order systems - This topic looks at some relatively simple first order systems, starting with Theories with Equality. Definition by Description. The Elementary Theory of the Natural Numbers (Peano's Axioms); results required later for Gödel's Theorem developed here. The Elementary Theory of Groups. Unbounded Dense Linear Orders as a theory which is complete yet not categorical.	Take-home assignment 2
5	Model theory 1 - Introduces the basic ideas associated with models of first order theories, structures and interpretations. This is sufficient to allow us to prove some useful things about the first order theories we will investigate.
6	Morse-Kelley (MK) Set Theory - The MK Axioms are used to set up Set Theory formally as a first order system. (MK is chosen in preference to the better-known ZF axiomatisation because it is more convenient for mathematics that deals extensively with classes; the ZF formulation is defined and the relationship between the two systems discussed in an appendix.) The major part of this chapter is devoted to showing how all of the standard techniques of Mathematics can indeed be developed from this formal basis. The Universe of Discourse and the Russell "paradox". Relations and functions as sets. Definition by induction. Well-ordering. The Axiom of Choice is discussed and several equivalent statements proved, including the Well-Order Principle, Zorn's Lemma and the Maximal Principal.	Take-home assignment 3
7	Transfinite arithmetic - Ordinal numbers, proofs and definitions by transfinite induction and ordinal arithmetic. The Cumulative Hierarchy. Cardinal numbers and cardinal arithmetic. The Schröder-Bernstein Theorem. Trichotomy for cardinal numbers is equivalent to the Axiom of Choice.
8	Model theory 2 - Adequacy of first order theories and the Löwenhein-Skolem Theorems. Models which respect equality. Compactness and some surprising results: a model of elementary arithmetic with an infinite number, a model of the reals with infinite numbers and infinitesimals.	Take-home assignment 4
9	Recursive functions - This topic and the next are covered because they are necessary for a proper understanding of the proofs in Chapter 11. Partial recursive, recursive and primitive recursive functions defined and the basic theory developed. Recursive sets and predicates.
10	Algorithms - A definition is given of a general algorithm; this involves the idea of a simple computer-type language. It is proved that a function is partial recursive if and only if it can be computed by a program in this language. A similar procedure for primitive recursive functions. With this background we can look at some more advanced topics in recursive functions. The Ackermann Function is recursive but not primitive recursive. An effective listing of the partial recursive functions. The s-m-n Theorem. Recursively enumerable sets. Important negative results (functions which cannot be computed) including the Halting Problem and Rice's Theorem.	Take-home assignment 5
11	Godel's Incompleteness Theorem - Discussed and proved
12	Appendices to the notes: (A) Construction of the basic number systems Further to Chapter 5, this shows how the Natural Numbers, the Integers, the Rationals, the Reals and the Complex Numbers can be built up from the axiomatic treatment. (B) Zermelo-Fraenkel (ZF) Set Theory - Counterpoint to Chapter 5, an outline of the ZF axioms for Set Theory and the differences between that approach and that of VBG. (C) Algorithms for Chapter 9 - The proof of Gödel's Theorem involves a number of assertions that certain functions are partial recursive or recursive. Detailed algorithms are provided here. (D) Turing machines - A brief description of this interesting topic, very relevant to computability, but not actually required for the development in these notes.
13	Extras - These notes are on subjects which are not part of the official syllabus. They have been supplied as quick introductions to subjects which are related to the course material and in which students in the past have expressed an interest. Nonstandard analysis - A (supposedly) easy way of doing calculus because it avoids the usual epsilon-delta type arguments. It follows fairly easily from the Compactness Theorem of Chapter 8 in the notes. Category theory - Another way of looking at mathematics? Independence of the Axiom of Choice - An important topic, central to the ideas of this course. A proof of this major result would be given as part of the course if it were not for the fact that it is so long and complicated there would be no room for anything else. But the proof, in all its gory detail, is here if you want to look at it.

Assessment Summary

Assessment task	Value	Due Date	Return of assessment	Learning Outcomes
Take-home assignment 1	16 %	15/03/2021	*	1,2,3
Take-home assignment 2	16 %	29/03/2021	*	1,2,3
Take-home assignment 3	16 %	19/04/2021	*	1,2,3
Take-home assignment 4	16 %	10/05/2021	*	1,2,3
Take-home assignment 5	16 %	24/05/2021	*	1,2,3
Final test	20 %	03/06/2021	01/07/2021	1,2,3

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information 
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the ANU Online website. In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Participation

Students are expected to attend at least one of the one-hour discussion meetings per week (see description of the course meetings below). When this is not possible students should access the audio recordings.

Examination(s)

There will be one two-hour formal examination in the final examination period.

It is a "hurdle" assessment, that is, it is necessary to pass this examination (achieve at least 50%) to pass the course.

It will be designed to be easy for anyone who has done the work throughout the course.

The date range in the Assessment Summary indicates the start of the end of semester exam period and the date official end of semester results are released on ISIS. Please check the ANU final Examination Timetable http://www.anu.edu.au/students/program-administration/assessments-exams/examination-timetable to confirm the date, time and location exam.

Assessment Task 1

Take-home assignment 1

Covers approximately Chapters 1–3.

The assignments consist usually of four questions of varying difficulty on the material. They are designed to involve approximately an evening's work, though this may vary of course depending on the student's background.

Approximately ten days are given from publication of the assignment to hand-in date, if possible including two weekends; the exact details will be decided in consultation with the class during the first two weeks of the course.

Assessment Task 2

Take-home assignment 2

Covers approximately Chapters 4–5.

The assignments consist usually of four questions of varying difficulty on the material. They are designed to involve approximately an evening's work, though this may vary of course depending on the student's background.

Approximately ten days are given from publication of the assignment to hand-in date, if possible including two weekends; the exact details will be decided in consultation with the class during the first two weeks of the course.

Assessment Task 3

Take-home assignment 3

Covers approximately the first half of Chapter 6.

The assignments consist usually of four questions of varying difficulty on the material. They are designed to involve approximately an evening's work, though this may vary of course depending on the student's background.

Approximately ten days are given from publication of the assignment to hand-in date, if possible including two weekends; the exact details will be decided in consultation with the class during the first two weeks of the course.

Assessment Task 4

Take-home assignment 4

Covers approximately the second half of Chapter 6 and Chapter 7.

The assignments consist usually of four questions of varying difficulty on the material. They are designed to involve approximately an evening's work, though this may vary of course depending on the student's background.

Approximately ten days are given from publication of the assignment to hand-in date, if possible including two weekends; the exact details will be decided in consultation with the class during the first two weeks of the course.

Assessment Task 5

Take-home assignment 5

Covers approximately Chapters 8–10.

The assignments consist usually of four questions of varying difficulty on the material. They are designed to involve approximately an evening's work, though this may vary of course depending on the student's background.

Approximately ten days are given from publication of the assignment to hand-in date, if possible including two weekends; the exact details will be decided in consultation with the class during the first two weeks of the course.

Assessment Task 6

Final test

The date range indicates the start of the end of semester exam period and the date official end of semester results are released on ISIS. Please check the course Wattle site and the ANU final Examination Timetable http://www.anu.edu.au/students/program-administration/assessments-exams/examination-timetable to confirm the date, time and location of the exam.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. At its heart, academic integrity is about behaving ethically, committing to honest and responsible scholarly practice and upholding these values with respect and fairness.

The ANU commits to assisting all members of our community to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to be familiar with the academic integrity principle and Academic Misconduct Rule, uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with.

The Academic Misconduct Rule is in place to promote academic integrity and manage academic misconduct. Very minor breaches of the academic integrity principle may result in a reduction of marks of up to 10% of the total marks available for the assessment. The ANU offers a number of online and in person services to assist students with their assignments, examinations, and other learning activities. Visit the Academic Skills website for more information about academic integrity, your responsibilities and for assistance with your assignments, writing skills and study.

Online Submission

You will be required to electronically sign a declaration as part of the submission of your assignment. Please keep a copy of the assignment for your records. We will not be using Turnitin.

Hardcopy Submission

We will not use hardcopy submission.

Late Submission

Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission of assessment tasks is not accepted after 10 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item. Late submission is not accepted for take-home examinations.

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

Assignments will be returned to students via e-mail from the marker/tutor.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

No

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students