Postgraduate Course: Logic, Computability and Incompleteness (PHIL11114)
Course Outline
School  School of Philosophy, Psychology and Language Sciences 
College  College of Arts, Humanities and Social Sciences 
Credit level (Normal year taken)  SCQF Level 11 (Postgraduate) 
Availability  Available to all students 
SCQF Credits  20 
ECTS Credits  10 
Summary  This course examines some fundamental topics relating to firstorder logic and the theory of computability, with particular emphasis on key limitative results. 
Course description 
This course will focus on key metatheoretical results linking computability and logic. In particular, Turing machines and their formalization in firstorder logic, linking uncomputability and the halting problem to undecidability of firstorder logic. We will then study recursive functions and their construction, followed by firstorder formalizations of arithmetic, particularly Robinson arithmetic and Peano arithmetic. We will then turn to the topic of the arithmetization of syntax and the diagonal lemma, before proceeding to prove some of the main limitative results concerning formal systems, in particular Godel's two incompleteness theorems, along with allied results employing the diagonal lemma, including Tarski's Theorem and Lob's Theorem.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 

Corequisites  
Prohibited Combinations  
Other requirements  None 
Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2023/24, Available to all students (SV1)

Quota: 11 
Course Start 
Semester 1 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
200
(
Lecture Hours 22,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
174 )

Assessment (Further Info) 
Written Exam
0 %,
Coursework
100 %,
Practical Exam
0 %

Additional Information (Assessment) 
30% midterm assignment and 70% final exam (take home) 
Feedback 
Guidance based on exercise sets assigned during the semester. 
No Exam Information 
Learning Outcomes
On completion of this course, the student will be able to:
 demonstrate analytical and abstract problem solving skills
 understand and engage with key limitative results in logic and computability theory, including the halting problem, the undecidability of firstorder logic, and the incompleteness of firstorder arithmetic
 grasp and analyze complex metatheoretical concepts
 formulate rigorous and precise pieces of logicomathematical reasoning.
 deploy rigorous formal methods including diagonalization and proof by mathematical induction.

Reading List
Core Syllabus Topics
Cardinality, Enumerability, Diagonalization
Turing Machines and Computability
Recursive Functions
FirstOrder Logic Revisited
Uncomputability and Undecidability
Completeness, Compactness and LowenheimSkolem
Formal Arithmetic
Diagonal Lemma, Godel and Tarski Theorems
Provability Predicates and Lob's Theorem
Recommended references:
[1] Boolos, G. & R. Jeffrey (1989) Computability and Logic. Cambridge University Press, 3rd edition.
[2] Machover, M (1996) Set Theory, Logic and Their Limitations. Cambridge University Press.
[3] Mendelson, E. (2015) An Introduction to Mathematical Logic. Chapman and Hall/CRC 6th edition
[4] Enderton, H. (2001) A Mathematical Introduction to Logic.
[5] Smith, P. (2013) An Introduction to Godel's Theorems. Cambridge University 
Additional Information
Graduate Attributes and Skills 
Ability to grasp and analyze complex theoretical concepts
Ability to utilize rigorous formal methods
Ability to analyse philosophical arguments
Ability to articulate and defend positions in a philosophical debate

Keywords  Turing machines,recursive functions,firstorder metatheory,Godel's theorems 
Contacts
Course organiser  Dr Paul Schweizer
Tel: (0131 6)50 2704
Email: paul@inf.ed.ac.uk 
Course secretary  Mrs Ida Conlin
Tel:
Email: iconlin@ed.ac.uk 

