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> >>> wrote: >>> >>>> Gödel used Principia Mathematica, and then a theory like PA can be shown >>>> essentially undecidable: adding axioms does not change incompleteness. >>>> That >>>> 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 >>>> of >>>> 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. > > > 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. >
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