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DRPS : Course Catalogue : School of Philosophy, Psychology and Language Sciences : Philosophy

Postgraduate Course: Logic, Computability and Incompleteness (PHIL11114)

Course Outline
SchoolSchool of Philosophy, Psychology and Language Sciences CollegeCollege of Arts, Humanities and Social Sciences
Credit level (Normal year taken)SCQF Level 11 (Postgraduate) AvailabilityAvailable to all students
SCQF Credits20 ECTS Credits10
SummaryThis course examines some fundamental topics relating to first-order 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 first-order logic, linking uncomputability and the halting problem to undecidability of first-order logic. We will then study recursive functions and their construction, followed by first-order 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)
Pre-requisites Co-requisites
Prohibited Combinations Other requirements None
Information for Visiting Students
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:
  1. demonstrate analytical and abstract problem solving skills
  2. understand and engage with key limitative results in logic and computability theory, including the halting problem, the undecidability of first-order logic, and the incompleteness of first-order arithmetic
  3. grasp and analyze complex metatheoretical concepts
  4. formulate rigorous and precise pieces of logico-mathematical reasoning.
  5. 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
First-Order Logic Revisited
Uncomputability and Undecidability
Completeness, Compactness and Lowenheim-Skolem
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
KeywordsTuring machines,recursive functions,first-order meta-theory,Godel's theorems
Course organiserDr Paul Schweizer
Tel: (0131 6)50 2704
Course secretaryMrs Ida Conlin
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