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Notation Here is a list of somewhat ideosyncratic symbols that will be used in this volume: f\A — restriction of a function / to a subset A of its domain f[A] — image of a set A under a function / A — symmetric difference of two sets a — abbreviation for (ao,... , a n ) K X — the set \JaKi a X of all functions from some ordinal a K into X \K — the cardinality of the set K A s""z — (where s G U) I and i G /) denotes the function s U {dom(s),i) TC(x) — the transitive closure of a set x, i.e., the smallest set y D x such that z C y for all z G y\ H\ — the family of all sets hereditarily of cardinality less than A, i.e., the family {x : \TC(x)\ A} [X]K — the family of all subsets of X of size K\ [X]K — the family of all subsets of X of size less than K Fin — denotes [o ] ^°, i.e., the family of finite subsets of CJ V — the set-theoretical universe, i.e., the class of all sets ON — the class of all ordinals LIM — the class of all limit ordinals Card — the class of all cardinals L — the constructible universe, i.e., the class of all constructible sets Va — the a-th level of the cumulative hierarchy, where V = UaeON ^* La — the a-th level of the constructible hierarchy, where L = IJaeON ^ (a, (3] — the set of ordinals {7 G O N : a 7 /3} OL-P — ordinal exponentiation is written with a dot in front of the exponent in order to distinguish it from cardinal exponentiation p.o. — abbreviates "partial order " l.o. — abbreviates "linear order " w.o. — abbreviates "wellorder " CH — abbreviates the Continuum Hypothesis GCH — abbreviates the generalized Continuum Hypothesis. Now let us review the rudiments of mathematical logic that were introduced in Chapter 5. The logical symbols of a first-order language L are A, -«, 3, =, brackets, and variable symbols V{ for every i G LJ. The symbols V, —, -•, V are considered abbreviations. Each language L also has nonlogical symbols: A set {r : i G / } of relational symbols, a set {fj : j G J} of functional symbols, and a set {ck : k G K} xi