Description: Define the cumulative hierarchy of sets function, using Takeuti and
Zaring's notation ( R1 ). Starting with the empty set, this
function builds up layers of sets where the next layer is the power set
of the previous layer (and the union of previous layers when the
argument is a limit ordinal). Using the Axiom of Regularity, we can
show that any set whatsoever belongs to one of the layers of this
hierarchy (see tz9.13 ). Our definition expresses Definition 9.9 of
TakeutiZaring p. 76 in a closed form, from which we derive the
recursive definition as theorems r10 , r1suc , and r1lim .
Theorem r1val1 shows a recursive definition that works for all values,
and theorems r1val2 and r1val3 show the value expressed in terms of
rank. Other notations for this function areR with the argument as a
subscript (Equation 3.1 of BellMachover p. 477), _V with a
subscript (Definition of Enderton p. 202),M with a subscript
(Definition 15.19 of Monk1 p. 113), the capital Greek letter psi
(Definition of Mendelson p. 281), and bold-face R (Definition 2.1 of
Kunen p. 95). (Contributed by NM, 2-Sep-2003)