On 4/20/2013 11:22 AM, WM wrote: > On 20 Apr., 17:30, fom <fomJ...@nyms.net> wrote: >> On 4/20/2013 3:16 AM, WM wrote: >> >>> Matheology § 255 >> >>> Let S = (1), (1, 2), (1, 2, 3), ... be a sequence of all finite >>> initial sets s_n = (1, 2, 3, ..., n) of natural numbers n. >> >>> Every natural number is in some term of S. >>> U s_n = |N >>> forall n exists i : n e s_i. >> >>> S is constructed by adding s_(i+1) after s_(i). >> >> Notice that WM is claiming that the sequence has >> a recursive definition. > > You can call it also inductive.
No. The domain of a recursive definition is an inductive set.
And, one can have a proof by induction whereby the form of a recursive definition facilitates the proof of the inductive step.
I am certain that one could find "inductive definition" in the literature. However, it confuses the relationships. One has inductive sets, inductive proofs, and recursive definitions.