Re: Goedel Numbers
My inclination would be to
define a relation called “hasGoedelNumber” which specifies the
number for that element. The relation is, of course, ternary being meaningful
only in a context, but how one handles that depends on how one handles contexts
in general. The range would be whole numbers.
For me an interesting
question would be whether such a relation would be a primitive concept or not,
and I think not. I would have a general relation “hasIndexNumber”
to handle all relations asserting a numbering for anything that one wants to
number, and the subrelations would be specialized by relating them to the
numbering system itself (again, suggesting an inherently ternary relation).
If one really wants a class of ‘Goedel Numbers”,
it would be the class of all numbers (a subset of integers) that are in the
range of that relation. Not particularly meaningful as a class, as you say.
These would be mental objects; the membership of the class would depend on
someone having defined a set of Goedel numbers for some purpose.
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher
Sent: Wednesday, February 02, 2011 12:17 PM
Subject: Re: [ontolog-forum] Where Do I Put...
On Feb 2, 2011, at 7:50 AM, John Bottoms wrote:
I've been thinking for a while that we need a "Where Do
thread. It will cover suggestions on where to put those fussy little
things that are not covered in ontological discussions. Do you have any
of these types of questions?
I recognize that there are many ways to organize an ontology, but
clearly if access is by structure, rather than by indexical, then we
should recognize and promote the more efficient structures.
Where Do I Put...??
1. Gõedel numbers and other formulas?
Puzzling question. First, Gödel numbers are not formulas;
they are, as the label suggests, numbers. Second, there are no Gödel
numbers per se. Rather, it only makes sense to call a number a
Gödel number relative to a Gödel numbering, i.e., a
completely conventional assignment of numbers to the basic syntactic elements
of a formal language. Gödel numberings, and the Gödel numbers they generate, are
purely theoretical constructs used for proving some rather advanced theorems in
mathematics (notably, of course, Gödel's incompleteness theorems).
I myself can't envision any scenario under which one would have to worry
about where to "put" them in the course of building an ontology
(whatever exactly "putting" is supposed to mean).
(...and are data encoded using Church's approach?)
What is Church's approach?
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