This course is a critical approach to the foundations of mathematics. In other mathematics classes, the philosophical concepts at the most basic foundations are usually treated naively. The question of what exactly a number is, or what a set or a proof or an algorithm are, is completely ignored. Some evidence that these matters are not insubstantial is that in the early twentieth century, naive attempts to address them by the great logicians of the time led to famous paradoxes and a period known as the Crisis in Foundations of Mathematics.

This course will address these matters directly and rigorously. It spends a few weeks on each of the following topics:

• First order logic
• Axiomatisation of set theory
• Model theory
• Computability
• Godel's Incompleteness Theorem
• Other topics in logic and set theory

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. Analyze and explain concepts from the foundations of mathematics.
4. Prove theorems in the foundations of mathematics.
5. Critically evaluate naive approaches to codifying the foundation of mathematics.

Examination Material or equipment

No permitted materials.

Required Resources

Notes on logic and set theory, by Peter Johnstone. Lecture notes for the course are linked to on Wattle.

Recommended Resources

There are a variety of online platforms you will use to participate in your study program at ANU, across all of your courses. These could include videos 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/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

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). Feedback can also be provided to Course Conveners and teachers via the Student Experience of Learning & Teaching (SELT) feedback program. SELT surveys are confidential and also provide the Colleges and ANU Executive with opportunities to recognise excellent teaching, and opportunities for improvement.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Content Block 1: First-order logic
2	Content Block 2: Recursive functions
3	Content Block 3: Axiomatic set theory
4	Content Block 4: Incompleteness theorems and other topics

Tutorial Registration

Workshop registration for this course is via MyTimetable. Please note that, due to student numbers, it is likely that only one workshop will be available. ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.

Assessment Summary

Assessment task	Value	Learning Outcomes
Pre-lecture questions	4 %	1,3,5
Workshop quizzes	16 %	1,2,3,4
Take-home assignments	50 %	1,2,3,4,5
Final exam	30 %	1,2,3,4,5

* 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 Integrity Rule before the commencement of their course. Other key policies and guidelines include:

• Academic Integrity Policy and Procedure
• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Guideline and General Information 
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure
• Code of practice for teaching and learning

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 Academic Skills 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

The lectures will be given in person in Psych G08. Recordings should be available on Echo360, which is accessible through Wattle. The purpose of the workshops is to discuss the material more informally with the demonstrator. Questions about the assignments are allowed.

Examination(s)

There will be one 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 so that meeting the hurdle will be straightforward for anyone who has done the work throughout the course.

Please check the course Wattle page 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 exam.

Assessment Task 1

Pre-lecture questions

Submit a question on the week's reading before each lecture, at least 10 out of 12 times. Each question is worth 0.4%, for a total of 4%.

Assessment Task 2

Workshop quizzes

One in each workshop. Each is worth 2%, the best 8 out of 11 taken, for a total of 16%. (The workshop in Week 8 falls on a holiday, so is not included in this assessment task. An optional replacement workshop on 26 April will be held.)

Assessment Task 3

Take-home assignments

There will be four assignments throughout the semester, each worth 12.5% towards the final grade.

• Assignment 1: due 12 March at 7am.
• Assignment 2: due 28 March at 5pm.
• Assignment 3: due 6 May at 7am.
• Assignment 4: due 24 May at 5pm.

Assessment Task 4

Final exam

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. The University’s students are an integral part of that community. The academic integrity principle commits all students to engage in academic work in ways that are consistent with, and actively support, academic integrity, and to uphold this commitment by behaving honestly, responsibly and ethically, and with respect and fairness, in scholarly practice.

The University expects all staff and students to be familiar with the academic integrity principle, the Academic Integrity Rule 2021, the Policy: Student Academic Integrity and Procedure: Student Academic Integrity, and to uphold high standards of academic integrity to ensure the quality and value of our qualifications.

The Academic Integrity Rule 2021 is a legal document that the University uses to promote academic integrity, and manage breaches of the academic integrity principle. The Policy and Procedure support the Rule by outlining overarching principles, responsibilities and processes. The Academic Integrity Rule 2021 commences on 1 December 2021 and applies to courses commencing on or after that date, as well as to research conduct occurring on or after that date. Prior to this, the Academic Misconduct Rule 2015 applies.

 

The University commits to assisting all students to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. All coursework students must complete the online Academic Integrity Module (Epigeum), and Higher Degree Research (HDR) students are required to complete research integrity training. The Academic Integrity website provides information about services available to assist students with their assignments, examinations and other learning activities, as well as understanding and upholding academic integrity.

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. This course does not use Turnitin, having been granted an exemption.

Hardcopy Submission

We will not use hardcopy submission.

Late Submission

• Late submissions not accepted: late pre-lecture questions, workshop quizzes, and take-home assignments, will not be accepted.

Referencing Requirements

The Academic Skills website has information to assist you with your writing and assessments. The website includes information about Academic Integrity including referencing requirements for different disciplines. There is also information on Plagiarism and different ways to use source material.

Returning Assignments

Assignments will be returned to students electronically.

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

Resubmission of assignments is not permitted.

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 Access 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