[FOM] A question concerning incompleteness

Panu Raatikainen panu.raatikainen at helsinki.fi
Thu Jun 5 01:07:03 EDT 2014

aa at tau.ac.il:

> Dear Fomers,
> I have a question:
> It is well-known that no consistent axiomatic extension
> of Robinson's system Q or Shoenfield's system N
> (from his great book "Mathematical Logic") can be complete.
> Does this theorem remain true if we delete from Q
> the axiom that states that every number different than 0
> has a predecessor, or if we delete from N the axiom
> that states that the relation < on the natural numbers
> is linear?
> Thanks,
> Arnon Avron

If I get the setting right, the answer is no:

Such a theory is what Smullyan (1961, 1993) has called "a Gödel
Theory": it is incomplete and undecidable; however, it is not
"a Rosser Theory", and hence, in Tarski’s terminology, not
essentially undecidable; that is, its every consistent extension
is not undecidable. For one may add to it, for example, the
sentence (For all x, x = 0) – or, any sentence that determines
the model’s size to be some finite cardinality – and the resulting theory is
complete and decidable.

I played a little with one such theory in a funny old paper:

(or here: http://www.mv.helsinki.fi/home/praatika/finitetruth.pdf)

There are references to some other uses as well (note 5).

All the Best

Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki

E-mail: panu.raatikainen at helsinki.fi


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