| Description: Define the cumulative
hierarchy of sets function, using Takeuti and
Zaring's notation (𝑅1). It
is also called the Von Neumann
hierarchy of sets function. 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 all previous layers when the argument
is a limit ordinal). Using the axiom of regularity, we can show that
every set belongs to one of the layers of this hierarchy (see
tz9.13 9781). Our definition expresses Definition 9.9 of
[TakeutiZaring]
p. 76 in a closed form, from which we derive the recursive definition as
Theorems r10 9758, r1suc 9760, and r1lim 9762. Theorem r1val1 9776 shows a
recursive definition that works for all values, and Theorems r1val2 9830
and r1val3 9831 show the value expressed in terms of rank.
Other notations for this function are R 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).
The values of this function, that is, the layers of this hierarchy, are
called "stages" to emphasize the process-like nature of this
function.
Some sources also use the term "level". (Contributed by NM,
2-Sep-2003.) |
