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Re: [ontolog-forum] Terminology and Knowledge Engineering

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Michael Brunnbauer <brunni@xxxxxxxxxxxx>
Date: Wed, 25 Jan 2012 00:48:09 +0100
Message-id: <20120124234809.GA14338@xxxxxxxxxxxx>

Hallo Rich,    (01)

some explanations, as I discovered that what I wrote is not Gödels first
incompleteness theorem but something that follows easily for first order
theories:    (02)

> On Tue, Jan 24, 2012 at 12:34:19PM -0800, Rich Cooper wrote:
> > Gödel showed that any logical system at least as
> > powerful as arithmetic is necessarily conflicted;
> > there are true theorems that cannot be proven and
> > there are false theorems that cannot be refuted.    (03)

This is the first incompleteness theorem: There will always be a theorem A
that cannot be proven and it's negation not A also cannot be proven.    (04)

As you can model arithmetic with a first order theory, Gödels completeness
theorem would be valid with such a theory. This means that if A and not A
cannot be proven, they also not entailed by the theory. This means the
negation of these statements:    (05)

 Every model of the theory is a model of A
 Every model of the theory is a model of not A    (06)

Which is equivalent to:    (07)

 There is a model of the theory that is not a model of A (where A is false)
 There is a model of the theory that is a model of A (where A is true)    (08)

> Gödel showed that there are always theorems that are independent of such a
> system. That is: They are neither true or false because there are models
> of the system where they are true and there are models where they are false.
> A model is an interpretation of all symbols of the system where all
> restrictions/axioms are true. The system is "ambiguous".    (09)

Regards,    (010)

Michael Brunnbauer    (011)

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