[FOM] inductive definitions in first order logic
rgheck
rgheck at brown.edu
Sat Apr 26 08:40:52 EDT 2008
Gergely Buday wrote:
> Dear FOMers,
>
> in a recent paper of Denecker I have found the following:
>
> "It is well-known that, in general, inductive definitions cannot be
> expressed in first-order logic (FO)."
>
> Could you please point me to the literature where this claim is
> argumented? And, what does "in general" means? This suggests that in
> special cases it is possible to include inductive definitions into
> predicate logic.
>
>
Since you don't say what the paper is, I can't know for sure what
Denecker has in mind. (You could ask him.) But here's a guess. First,
"in general" means: Not all inductive definitions can be expressed in
FOL. Second, by "expressing an inductive definition", I take it that
what he has in mind is something that is semantical. Of course, in PA,
we can get the effect of certain sorts of inductive definitions, via
coding by finite sequences and recursion on the course of values. But,
this will not give you the right result semantically, because of the
existence of non-standard models. Now, back to "in general": Of course,
some inductive definitions can be expressed in first-order PA in that
sense, because some are completely trivial, e.g.:
f(0) = 0
f(n + 1) = 0
No problem there!
Richard Heck
--
-----------------------
Richard G Heck Jr
Professor of Philosophy
Brown University
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