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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