Unit of study_

# MATH3066: Algebra and Logic

This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times. Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. A range of applications may be presented, in particular, impossibility arguments such as the unsolvability of the celebrated classical construction problems of the Greeks. Quotient rings are introduced, culminating in a construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of null sequences. Axiomatics are placed in the context of reasoning within first order logic and set theory. The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including sketches of proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem and the undecidability of First Order Logic.

Code MATH3066 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? 6 credit points of MATH2XXX None MATH3062 or MATH3065 Introductory knowledge of group theory. For example as in MATH2X22

At the completion of this unit, you should be able to:

• LO1. be fluent in analysing and constructing logical arguments
• LO2. be conversant with the Propositional and Predicate Calculi, and related notions of syntax (deduction) and semantics (completeness)
• LO3. be informed about the historical underpinnings of abstract algebra that lead to axiomatic theories of groups, rings, integral domains and fields, and their use in exploring questions of decidability or impossibility
• LO4. be fluent with a range of standard and exotic arithmetics and ring constructions, and be able to prove elementary propositions and theorems about them
• LO5. understand the Halting Problem and Turing’s use of it to prove the undecidability of first order logic.

## Unit outlines

Unit outlines will be available 1 week before the first day of teaching for the relevant session.