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>wrote:GĂ¶del used Principia Mathematica, and then a theory like PA can beshownessentially undecidable: adding axioms does not changeincompleteness. Thatis why it applies to us, as far as we are correct. It does notapply toeveryday reasoning, as this use a non monotonical theory, with anotion ofupdating our beliefs.Not all undecidable theory are essentially undecidable. Grouptheory isundecidable, but abelian group theory is decidable.Bruno, is there an general, meta-mathematical theory about whataxioms 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.`

Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.