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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