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
the set \JaKi
of all functions from some ordinal a K into X;
the cardinality of the set
s""z (where s G
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};
the family of all subsets of X of size 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}
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