`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.

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.

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/