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Definition df-irdg 6311
 Description: Define a recursive definition generator on (the class of ordinal numbers) with characteristic function and initial value . 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 operation (especially when df-recs 6246 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of . This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication. For finite recursion we also define df-frec 6332 and for suitable characteristic functions df-frec 6332 yields the same result as restricted to , as seen at frecrdg 6349. Note: We introduce with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.)
Assertion
Ref Expression
df-irdg recs
Distinct variable groups:   ,,   ,,

Detailed syntax breakdown of Definition df-irdg
StepHypRef Expression
1 cF . . 3
2 cI . . 3
31, 2crdg 6310 . 2
4 vg . . . 4
5 cvv 2712 . . . 4
6 vx . . . . . 6
74cv 1334 . . . . . . 7
87cdm 4583 . . . . . 6
96cv 1334 . . . . . . . 8
109, 7cfv 5167 . . . . . . 7
1110, 1cfv 5167 . . . . . 6
126, 8, 11ciun 3849 . . . . 5
132, 12cun 3100 . . . 4
144, 5, 13cmpt 4025 . . 3
1514crecs 6245 . 2 recs
163, 15wceq 1335 1 recs
 Colors of variables: wff set class This definition is referenced by:  rdgeq1  6312  rdgeq2  6313  rdgfun  6314  rdgexggg  6318  rdgifnon  6320  rdgifnon2  6321  rdgivallem  6322  rdgon  6327  rdg0  6328
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