From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3310 Path: news.gmane.org!not-for-mail From: "Marta Bunge" Newsgroups: gmane.science.mathematics.categories Subject: RE: Construction of a real closure Date: Sun, 07 May 2006 16:28:38 -0400 Message-ID: <39718.6017282822$1241019221@news.gmane.org> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed X-Trace: ger.gmane.org 1241019221 8196 80.91.229.2 (29 Apr 2009 15:33:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:33:41 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon May 8 10:15:33 2006 -0300 X-Keywords: X-UID: 230 Original-Lines: 59 Xref: news.gmane.org gmane.science.mathematics.categories:3310 Archived-At: Dear Michael, >Incidentally, AC doesn't bother me at all, but constructive methods are >interesting in their own right. Constructive methods are sometimes more than "just interesting in their own right". If a certain theorem expressible in geometric logic can be proved constructibly, then it is true in any topos, and so amenable to a variety of interesting interpretations. I return to one of the examples I mentioned from my paper ("Sheaves and prime model extensions", J. Algebra 68, 1981) to illustrate how it works for the real closure, which has both a constructive and a non-constructive part. Preamble. The following theorem has been shown (Lipshitz, Trans AMS 211, 1975; van der Dries Ann. Math. 12, 1977) in order to solve the analogue of Hilbert's 17th problem for commutative (von Neumann) regular f-rings. Theorem. Any commutative regular f-ring has an atomless real closure. Using the Pierce representation of any commutative (von Neumann) regular f-ring R by an ordered field K in the (non-Boolean) topos E of sheaves on the spectrum of K, the above theorem can be shown more easily than in the above quoted sources, simply using the analogue for ordered fields, with some care, since we know that, although there is an algorithm for constructing the "real closure" of an ordered field, this does not automatically give that such is real closed. This is how I proceed, using the "mid-way house" method. The inclusion F = E_{not not} >---> E factors through the Gleason cover g:G--->E (g a surjection) via a flat inclusion i: F >---> G. Since F is a BVM/ST, i^*g^*K >---> K' has a real closure. Thus, since i_* preserves finitary logic, g^*K >---> i_*K' is an extension of g^*K into a real closed field i_*K'. Sturm's theorem gives an algorithm which makes sense in any topos, provided the ordered field one applies it to is already contained in some real closed field. From this follows first that g^*K has a real closure in G, and then, by the surjectivity of g (g^* faithful), follows that K has a real closure in E. By "transfer", the commutative regular f-ring R ( R = \Gamma (K)) has a real closure (in Sets). Other results (including classically new ones) in the case of differential rings are proved in that paper. Sheaf methods in the theory of commutative rings are very useful in classical mathematics. See for instance "Recent Advances in the Representation Theory of Rings and C*-Algebras by Continuous Sections", Memoirs AMS 148, 1973. In particular, C. Mulvey, "Intuitionistic Algebra and Representation of Rings" in that volume). Another source is "Applications of Sheaves" (Proceedings Durham 1977), LNM 753, Springer 1977. In particular, the papers by Fourman and Hyland, Fourman and D. Scott, and C. Rousseau. Anyway, this is a huge subject and hints at the usefulness of constructive methods in algebra and analysis, when available. Anhyway, trhis is old stuff. Best, Marta