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Herbert Enderton in his A Mathematical Introduction to Logic 2nd edition, proves a theorem (a "recursion theorem") in section 1.4, p. 39.

Using his example, the idea is the following: We have some set $C \subseteq U $ generated "recursively" from some subset $B\subseteq C$, using some functions: $\ f:U\times U\rightarrow U\ $and $\ g:U\rightarrow U\ $ (in this case meaning simply that $C$ is closed under $f$ and $g$).

(For example $C$ might be the set of formulas of propositional logic generated from $B$, the atomic formulas, using formula building operations $f$, $g$, etc.)

The theorem shows that when a function $\ \bar{h} :C\rightarrow V\ $ is defined (here $V$ is just some set), using functions
$h:B\rightarrow V\ $
$ F:V\times V\rightarrow V\ $
$ G:V\rightarrow V\ $

so that:
(i) For $x$ in $B$: $\ \bar{h}(x)=h(x)$
(ii)For $x, y$ in $C$:
$ \bar{h}(f(x,y))=F(\bar{h}(x),\bar{h}(y))$
$ \bar{h}(g(x))=G(\bar{h}(x))$

then under certain condition (i.e. when: $f$ and $g$ are one-to-one, and when their ranges and $B$ are pairwise disjoint), there exists a unique such function $\bar{h}$ on $C$ meeting the given requirements (i)&(ii) (see p. 39 for a precise statement of the theorem).

(So for example $\bar{h}$ might be a truth assignment function for the formulas of propositional logic.)

As Enderton himself implies, the theorem would hold if instead of $f$ and $g$, we would have some functions $f_{1},f_{2},..,f_{n}$ and corresponding rules for $\bar{h}$. But he only proves the theorem for the special case mentioned here.

My question is: Does the theorem generalize to a case where $C$ is generated by a countably infinite set of functions $f_{1},f_{2},...$ , and so we also have infinitely many rules for computing $\bar{h}$ ?

I assume it does generalize, but I'm looking for a reference. Does anyone have a reference where the general case is proved?

Johannes
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  • There are a couple of related references mentioned here but I don't know if any of them contain exactly what you want. http://math.stackexchange.com/questions/900685/readings-on-more-general-abstract-notions-of-induction-related-to-logic – Ishfaaq May 22 '15 at 00:22
  • @Ishfaaq The "Handbook of Mathematical Induction" looks most promising, because the others look too advanced for this simple (?) theorem, I did skim them. Unfortunately I have no access to that book. I need just the reference for a term paper, I don't really have space in the paper to include the actual proof. And anyway it's not a math paper, so I try to avoid such things as much as possible. – Johannes May 22 '15 at 01:49

1 Answers1

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This is a bit of a hack, but you might reinterpret the many functions $f_i:U\times U\to U$ as a single function $f:\omega\times U\times \omega\times U\to \{i\}\times U$, and correspondingly the many rules $h_i$ on $U$ as a single rule on $\omega\times U$.

mmw
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  • Thanks, I'm thinking that because C is generated in "stages" using the $f1, f2..$ functions, the simplest proof is probably some kind of induction on these stages of generating $ \bar{h} $. But I'm really after a reference. – Johannes May 21 '15 at 15:47
  • ...What I gave you is a purported proof! But if you want a reference, check out Aczel's "An introduction to inductive definitions" in Barwise et al, Handbook of Mathematical Logic. – mmw May 21 '15 at 16:04