User antongiulio - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:27:34Z http://mathoverflow.net/feeds/user/6988 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46970/proofs-of-the-uncountability-of-the-reals/47425#47425 Answer by Antongiulio for Proofs of the uncountability of the reals. Antongiulio 2010-11-26T13:26:00Z 2010-11-26T13:26:00Z <p>I have the following candidate: <a href="http://arxiv.org/abs/1003.3557" rel="nofollow">http://arxiv.org/abs/1003.3557</a>, section 7.4. Notice that in the setting of the article one cannot use diagonalization.</p> http://mathoverflow.net/questions/41573/proofs-of-baire-category-theorem Proofs of Baire category theorem Antongiulio 2010-10-09T09:23:48Z 2010-10-09T21:45:38Z <p>I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere).</p> <p>My motivation is the following: in the article <a href="http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/papers/117/index.php" rel="nofollow">Definably complete and Baire structures</a> we defined a first-order notion of Baire structures, and I would like to prove that every definably complete ordered field is definably Baire. To do that, a possible approach would be to take a proof of the fact that \$\mathbb R\$ is not meager, and adapt it to the first-order situation. The main obstacle to such an adaptation is the fact that we cannot define sets by recursion.</p> http://mathoverflow.net/questions/38324/did-pogorzelski-claim-to-have-a-proof-of-goldbachs-conjecture/38327#38327 Answer by Antongiulio for Did Pogorzelski claim to have a proof of Goldbach's Conjecture? Antongiulio 2010-09-10T16:14:35Z 2010-09-10T16:14:35Z <p>"The procedure by which proofs become accepted as essentially correct is by publication in a journal after peer review": that's false. Many published proofs contain substantial mistakes. Nobody should accept a result only because it has been published.</p> http://mathoverflow.net/questions/29100/real-algebraic-geometry-vs-algebraic-geometry/29112#29112 Answer by Antongiulio for Real algebraic geometry vs. algebraic geometry Antongiulio 2010-06-22T16:24:17Z 2010-06-22T16:24:17Z <p>For an easy introduction to RAG, you could read van den Dries book "Tame topology and o-minimal structures": he treats the more general notion of o-minimal structures instead of real closed fields, and he does not uses any tool from AG.</p> http://mathoverflow.net/questions/48732/can-we-alter-the-axioms-of-euclidean-space-to-have-mathbbq3-as-a-unique-mod/48754#48754 Comment by Antongiulio Antongiulio 2010-12-09T13:10:56Z 2010-12-09T13:10:56Z From the wikipedia article: &quot;Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic&quot;. On the other hand, if you use first-order axioms, no infinite field (and no theory of a projective plane that can define a field, i.e. a Desarguesian one: see &lt;a&gt;<a href="http://en.wikipedia.org/wiki/Projective_plane&lt;/a&gt" rel="nofollow">en.wikipedia.org/wiki/Projective_plane&lt;/a&gt</a>;) can be omega-categorical. http://mathoverflow.net/questions/41573/proofs-of-baire-category-theorem Comment by Antongiulio Antongiulio 2010-10-11T10:05:34Z 2010-10-11T10:05:34Z @gowers: there are at least two different proofs of the fact that R is not meager (corresponding to the two cases of Baire category theorem): one using the fact that R is a complete metric space, the other using the fact that [0,1] is compact.