MATH3066 Algebra And Logic Notes | Assignment Help | Syllabus
University: University Of Sydney
“This unit of study integrates and expands mathematical concepts and methods that the majority of participants will have encountered in their first and second years, and will be of broad interest to all pure and applied mathematics students. It uses algebra and logic to present and solve a series of linked problems that have been important to the evolution of mathematics from antiquity to the present day. The Propositional and Predicate Calculi are examined separately as model axiomatic systems, with demonstrations of consistency and completeness. The course concludes with a discussion of Godel’s Incompleteness Theorem, which introduces precise notions of computability and decidability through abstract Turing machines, culminating in the unsolvability of the Halting Problem, the undecidability of First Order Logic, and the unsolvability of the Halting Problem. Field and ring axioms, as well as the language of field extensions, are used to introduce, unify, and explain abstractly classical and new arithmetics. Quotient rings, which are utilized to build various finite and infinite fields, are presented. The real numbers are constructed by factoring out rings of Cauchy sequences of rationals by the ideal of sequences. Within first order logic and set theory, axiomatics is used to explain reasoning.
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