Description: Define a recursive
definition generator on (the class of ordinal
numbers) with characteristic function and initial value .
This combines functions in tfr16815
and in tz7.44-16821 into one
definition. This rather amazing operation allows us to define, with
compact direct definitions, functions that are usually defined in
textbooks only with indirect self-referencing recursive definitions. A
recursive definition requires advanced metalogic to justify - in
particular, eliminating a recursive definition is very difficult and
often not even shown in textbooks. On the other hand, the elimination
of a direct definition is a matter of simple mechanical substitution.
The price paid is the daunting complexity of our rec operation
(especially when df-recs6791 that it is built on is also eliminated). But
once we get past this hurdle, definitions that would otherwise be
recursive become relatively simple, as in for example oav6912,
from which
we prove the recursive textbook definition as theorems oa06917,
oasuc6925,
and oalim6933 (with the help of theorems rdg06836,
rdgsuc6839, and
rdglim2a6848). We can also restrict the rec operation to define
otherwise recursive functions on the natural numbers ; see
fr0g6850 and frsuc6851. Our rec operation apparently does not appear
in published literature, although closely related is Definition 25.2 of
[Quine] p. 177, which he uses to
"turn...a recursion into a genuine or
direct definition" (p. 174). Note that the if operations (see
df-if3769) select cases based on whether the domain of
is zero, a
successor, or a limit ordinal.
An important use of this definition is in the recursive sequence
generator df-seq11748 on the natural numbers (as a subset of the
complex
numbers), allowing us to define, with direct definitions, recursive
infinite sequences such as the factorial function df-fac11993 and integer
powers df-exp11807.
Note: We introducerecwith
the philosophical goal of beingable to eliminate all definitions with direct mechanical
substitutionand to verify easily the soundness of definitions. Metamath
itselfhas no built-in technical limitation that prevents multiple-partrecursive definitions in the traditional textbook style.
(Contributed
by NM, 9-Apr-1995.) (Revised by Mario Carneiro,
9-May-2015.)