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
);
KX
— the set \JaKi
aX
of all functions from some ordinal a K into X;
\K
— the cardinality of the set
KA;
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}
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