Re: Universal grammar

In article <haberg-2510062309000001@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hans Aberg <haberg@xxxxxxxxxx> wrote:
In article <ehoiel$5q5a@xxxxxxxxxxxxxxxxxxxx>, hrubin@xxxxxxxxxxxxxxxxxxxx
(Herman Rubin) wrote:

Working math usesnaive set theory, and it is rare to indicate which
formal axiomatic version one uses.
Theconstructioncorrespondprogrammingwith respect to interfaces.

Many mathematicians use naive set theory, but the field
does not. If the paradoxes of naive set theory arise,
they will be quickly made known.

Let's take an example: The axioms for natural numbers are well sufficient
for doing number theory. If one would find that a numbertheoretic result
would depend on itsconstruction in axiomatic settheory (like 0 = empty
set, n = {0, 1, ..., n - 1}), it is probably something wrong with the
construction of the natural numbers.

The categorization from the Peano Postulates shows that
this cannot be the case.

One would then not use the Peano axioms.

One has to use something of that sort to get the power of the
usual integers. The set of axioms to take for a system is a
subset of the set of theorems adequate to get the rest. So
if one has the integers, the Peano Postulates have to be
provable in the system.

In fact, the axiomatic set theory
VERSION is not the only way ordinal numbers can be handled;
they can easily be handled in set theory without a
canonical representation, which is the case with cardinal
numbers in some models. Ordinals were used in set theory
in the 19th century, and correctly; the von Neumann model,
which you have used, goes back to von Neumann's thesis,
in the 1920's. The finding of canonical models does not
mean they must be used; nobody uses Dedekind's models of
the integers, in which n+1 = {n}.

This is what I try to say: The natural numbers act as an interface, whose
exact implementaion is irrelevant in working math.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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