On Thu, Dec 13, 2012 at 6:26 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:
> On 12 Dec 2012, at 23:39, meekerdb wrote:
>> On 12/12/2012 7:27 AM, Richard Ruquist wrote:
>>> On Tue, Dec 11, 2012 at 10:08 AM, Bruno Marchal<marc...@ulb.ac.be>
>>>> Gödel used Principia Mathematica, and then a theory like PA can be shown
>>>> essentially undecidable: adding axioms does not change incompleteness.
>>>> is why it applies to us, as far as we are correct. It does not apply to
>>>> everyday reasoning, as this use a non monotonical theory, with a notion
>>>> updating our beliefs.
>>>> Not all undecidable theory are essentially undecidable. Group theory is
>>>> undecidable, but abelian group theory is decidable.
>> Bruno, is there an general, meta-mathematical theory about what axioms
>> will produce a decidable theory and which will not?
> I have never heard about a simple recipe. The decidability of the theory of
> abelian group has been shown by Wanda Szmielew ("Arithmetical properties of
> Abelian Groups". Doctoral dissertation, University of California, 1950), see
> also "Decision problem in group theory", Proceedings of the tenth
> International Congress of Philosophy, Amsterdam 1948).
> The undecidability of the elementary theory of group is proved by Tarski,
> and you can find it in the book (now Dover) Undecidable Theories, 2010.
> Tarski has also proved the decidability of the elementary (first order)
> theory of the reals (with the consequence that you cannot define the natural
> numbers from the reals with the real + and * laws).
> Natural numbers are logically more complex than the real numbers.
> Same with polynomial equations: undecidable with integers coefficients and
> unknown, but decidable on the reals.
> In the real, adding the trigonometric functions makes possible to define the
> natural numbers (by sinPIx = 0), and so the trigonometric functions
> reintroduce the undecidability.
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