The Fibonacci sequence, taking the nickname of creator Leonardo Bonacci, has a simple,

recursive definition: take one as the first term, one as the second term, and then find each subsequent term by adding the two preceding terms.

[a.sub.l] = [a.sup.(q).sub.l] has the following

recursive definition:

Furthermore, the order relation encodes information so that each step of the procedure computation, modeled by each iteration of the nonrecursive mapping, is identified with a model element that is greater than (or equal to) the elements representing the preceding steps of the procedure computation (the preceding iterations of the nonrecursive mapping), since each iteration gives more information about the meaning of the

recursive definition of the procedure than all previous ones.

Dave Morice commented on the

recursive definition of WORD: At the fourth level, the meaning of WORD verges on magical.

Imagine, for example, that we simply translated the

recursive definition of theoremhood for some axiom system for classical logic into a correspondingly

recursive definition on the model-theoretic level, and then restate and prove the completeness theorem for classical logic in terms of that translation.

Its type theory is a scaffolding built into its use of structured variables--a scaffolding which emulates the fundamental laws of a type- and order-regimented theory of attributes and thereby emulates a type-theory of classes and relations-in-extension." He further holds that Principia offers a

recursive definition of "truth," which assumes that the facts are truth-markers, and the universals inhering in them are logically independent of each other.

Thus, the Euler number [sub.[CHI]([[bar.x].sup.c] of its topological closure may be defined by Hadwiger's

recursive definition (Schneider, 1993, p.

Though proving that the resulting collection is a basis with symmetric Hilbert series is better done from the

recursive definition, the construction is better motivated from this viewpoint.

Now, for each Fibonacci sequence we have investigated above, we have been increasing the number of previous terms to be added in the

recursive definition. Let us consider a slight variation of this.

Tarski provided us with the first precise formulation of the T-schema, while also showing how to give a

recursive definition of truth.

is a discrete

recursive definition of [F.sub.n] iff D [is greater than or equal to] 1; [R.sub.d,n] = [w.sub.d] + [r.sub.d,n] [is greater than or equal to] O, where [w.sub.d] [is greater than] 0 and [[Sigma].sub.1 [is less than or equal to] d [is less than or equal to] D] |[r.sub.d,n]| = O([n.sup.-[Rho]]) for some [Rho] [is greater than] 0; and [S.sub.d,n] = [z.sub.d] [multiplied by] n + [s.sub.d,n], where 0 [is less than] [z.sub.d] [is less than] 1 and [[Sigma].sub.1 [is less than or equal to] d [is less than or equal to] D] |[s.sub.d,n]| = O ([n.sup.-[Sigma]]) for some [Sigma] [is greater than] 0.

Following again the lead of Levinson, all this can be usefully put in some kind of

recursive definition. Over-simplified, the definition is: