Intermediate set theory
The authors cover first order logic and the main topics of set theory in a clear mathematical style with sensible philosophical discussion. The emphasis is on presenting the use of set theory in various areas of mathematics, with particular attention paid to introducing axiomatic set theory, showing how the axioms are needed in mathematical practice and how they arise. Other areas introduced include the axiom of choice, filters and ideals. Exercises are provided which are suitable for both beginning students and degree-level students.
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Some of the history of the concept of sets
Firstorder logic and its use in set theory
The axioms of set theory
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absolute abstraction terms antichain assignment assume atomic formulas axiom of choice axiom of extensionality axiom of foundation axiom of infinity bijection Boolean algebra bound Cantor cardinal numbers chain choice function cofinality collection consider construction continuum hypothesis contradiction countable cumulative type structure Dedekind define definition delta-system denote dense domain empty set exercise extend fact filter finite sets first-order logic forcing formal free variables give given Godel's hereditarily symmetric hold ideal implies induction hypothesis initial ordinal integers isomorphic language lattice limit ordinal mathematics model of ZF natural numbers non-empty Note notion paradox partial ordering Peano poset power-set axiom proof proper class properties prove q lh quantifiers rank rank(x real numbers recursion theorem replacement axiom restrict result satisfy set theory Skolem functions subgroups subset axiom sum-set suppose symbols transfinite induction transitive truth lemma uncountable well-ordering Zorn's lemma