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.


Reply via email to