Re: An example of a complete but undecidable theory

From: Mike Oliver (moliver_at_unt.edu)
Date: 08/26/04 

Date: Thu, 26 Aug 2004 12:54:57 -0500

"H. Enderton" wrote:
> Mike Oliver <mike_lists@verizon.net> wrote:
>> So here's a question to which I don't know the answer off
>> the top of my head: If T is decidable, is there a theory
>> T' in some other language, such that T and T' are mutually
>> relatively interpretable, and T' is complete? If so,
>> then you might argue that a decidable theory is "morally"
>> complete, except that its language is too rich, has too
>> many symbols that T doesn't say enough about.
>
> Let's take T to be the theory of equality, i.e., the set of
> valid sentences in the language {=}. T is decidable, but
> not complete.

Well, this certainly doesn't seem to be a case of the language
being too rich. Let's see, an example of a sentence independent
of T would be something like "there are two distinct objects".
What if we restrict to theories without finite models?

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