At 20:08 07/12/04 -0500, Hal Ruhl wrote:
>I believe we discussed this and you agreed that a complete arithmetic
>would be inconsistent. I have not found the applicable posts.
If by arithmetic you mean an axiomatizable theory, then indeed, by
incompleteness it follows that such an arithmetic, if complete, must be
inconsistent.
If by arithmetic you mean a (not necessarily axiomatizable, and actually:
necessarily not axiomatizable) model, then incompleteness does not apply. A
model (identified with some set of sentences) can be both complete and
consistent.
Sometimes people use "arithmetic" (with a little "a") for an axiomatizable
presentation of arithmetic, and Arithmetic for the set of sentence true in
the "standard model" of arithmetic.
>We have reached too many levels of nesting. I have been of on my own
>excavations. Is not "all true arithmetical sentences" a part of comp?
"Comp" just asks for the truth of those sentences not depending of me or you.
My problem is that I have not a clear idea of what you mean by nothing,
dynamic, boundary, all.
About the inconsistency of the "ALL" I could imagine a resemblance with my
critics of Tegmark, which is that if you take a too bigger mathematical
ontology you take the risk of being inconsistent (i.e. that your theory is
inconsistent).
It is like giving a name to the unnameable.
Before axiomatic set theories like Zermelo-Fraenkel, ... Cantor called the
"collection" of all sets the "Inconsistent". But this does make sense for
me. Only a theory, or a machine, or a person can be inconsistent, not a
set, or a realm, or a model.
Bruno
http://iridia.ulb.ac.be/~marchal/
Received on Wed Dec 08 2004 - 10:30:53 PST