# Thread: Recursion, short question

1. ## Recursion, short question

The recursion theorem

In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set X, an element a of X and a function , the theorem states that there is a unique function (where denotes the set of natural numbers including zero) such that

for any natural number n.
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Given a standard recitation of the "Recursion Theorem" is there a proper name for what is labeled the function "f" in the above statement, the f of f(F(n)).

If I were naming it I would call it the "Recursor function" as an analog of the "successor function" but as yet nobody has named me king. Is there is proper name for that function so that it can be talked about without the need to recite the Recursion theorem to make a specific reference.

That's it.

2. ## Re: Recursion, short question

I've never heard of one. Probably because it can be ANY function. The function we are DEFINING recursively is $F$, often, in a recursive definition the function $f$ isn't even called a function, but a "formula" or "expression". Recursor sounds as good a name as any.

3. ## Re: Recursion, short question

Thanks. In now re-reading earlier posts from a slightly higher elevation I see many points and implications which, although stated clearly, just didn't "sink in", leaving me both appreciate of what I have learned and sort of cringing over many simpleton misunderstandings. But, ok, the nature of the beast. I am in the process of consolidating notes, my feeling is that at last I am on playing board with regard to PMI. I appreciate everyone's insights.