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Elementary Operations on Sets
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arithmetic assume Axiom of Choice Axiom of Constructibility binary relation Borel sets called Chapter choice function closed sets closed unbounded computation Continuum Hypothesis contradiction countable set deﬁned Deﬁnition Deﬁnition Let denote dense linearly ordered equipotent equivalence example Exercises 3.1 exists ﬁlter ﬁnd ﬁnite sequence ﬁnite sets ﬁrst follows function f greatest element hence Hint holds implies inaccessible cardinal Induction Principle inﬁnite sequence inﬁnite set initial ordinal isomorphic K-complete large cardinals least element Let f Let oz limit ordinal linear linearly ordered set mathematical mutually disjoint natural numbers nonprincipal notation one-to-one mapping open intervals operation ordered pairs ordinal number ordinal oz partition perfect subset proof of Theorem prove rational numbers real numbers Recursion Theorem satisﬁes set theory sets of reals supremum system of sets Theorem Let Transﬁnite Recursion transitive ultraﬁlter uncountable union unique upper bound well-ordered set