A set of introductory notes on topos theory, which is an abstract theory of generalized sets (though toposes, particularly Grothendieck toposes, can also be regarded as generalized topological spaces). After a brief review of some basic concepts from category theory, toposes are defined and some simple examples are given. The category of sets is described and characterized as the topos of constant sets. Topos logic is then described from the internal and external perspectives and then the internal logic of a topos is presented in the language of local set theories. The notes conclude with a consideration of number systems and arithmetic in the context of topos theory.
A set of introductory notes on axiomatic set theory presented in an informal way (which could be called "naive axiomatic set theory", since it is informal as naive set theory is but also presented in terms of axioms). The sections included are: fundamentals of sets and classes; products and coproducts; functions; relations; quotients; finite and infinite sets; well-order and induction; ordinals and cardinals; the axiom of choice; and the number systems (from the natural numbers to the real and complex numbers).
The first part of a basic introduction to algebraic Galois theory. After a summary of the basic concepts and definitions of group theory and field theory, the basics of field extensions and some basic results from group theory are presented. This part concludes with a statement of the beautiful fundamental theorem.
The second part of a basic introduction to algebraic Galois theory. After a review of the concepts and definitions from Part I, a galois extension is defined and several examples are given, followed by an illustration of the fundamental theorem. The notes conclude with a more complete version of the beautiful fundamental theorem.