User ian maxwell - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:34:58Z http://mathoverflow.net/feeds/user/1448 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108588/axiomatization-of-the-incidence-geometry-of-the-euclidean-plane Axiomatization of the incidence geometry of the Euclidean plane Ian Maxwell 2012-10-02T01:31:30Z 2012-10-02T01:31:30Z <p>There are several well-known axiomatizations of Euclidean plane geometry, the language of which is usually considered to include at least the relations of</p> <ul> <li>incidence (point-line, point-segment, or possibly point-circle),</li> <li>betweenness (point-point-point), and</li> <li>congruence (segment-segment).</li> </ul> <p>(One could define incidence in terms of betweenness---Tarski does this, for example---but I'm interested in the case where these are distinct primitives.)</p> <p>These axiomatizations are complete, so <em>a fortiori</em> they generate a complete theory in the language of incidence geometry. But on the face of it, there may be theorems of Euclidean plane geometry whose statements are in the language of incidence geometry, but which cannot be proven without appeal to axioms involving betweenness or congruence.</p> <blockquote> <p>Are there in fact any such theorems?</p> </blockquote> <p>And if so,</p> <blockquote> <p>Is there a known 'nice' axiomatization of the incidence geometry of the Euclidean plane, in the language of incidence geometry alone?</p> </blockquote> <p>I'm not sure how to formalize what I mean by 'nice.' Essentially I want to exclude such answers as "the set of all theorems of Euclidean geometry in the language of incidence geometry," which is a recursively enumerable set but seems like a bit of a cop-out.</p> http://mathoverflow.net/questions/12905/set-theories-that-do-require-the-existence-of-urelements/57259#57259 Answer by Ian Maxwell for Set theories that do require the existence of urelements? Ian Maxwell 2011-03-03T15:52:02Z 2011-03-03T15:52:02Z <p>I don't know if this is exactly what you're looking for, but it's a theorem of NFU that $|\mathcal{P}(V)| &lt; |V|$---which has as a corollary not only that there are atoms, but that the set of atoms is equipollent with the universe.</p> <p>(A somewhat more disquieting way of putting this is that there are more atoms than there are sets.)</p> http://mathoverflow.net/questions/14330/decidability-of-the-axiom-of-choice/56432#56432 Answer by Ian Maxwell for Decidability of the Axiom of Choice Ian Maxwell 2011-02-23T19:15:47Z 2011-02-23T19:15:47Z <p>There is an interesting question here: "Are there any set theories in which the question of choice is decided <em>without</em> inserting an axiom tailored to the purpose?"</p> <p>As Jeremy points out, NF is inconsistent with choice. It's strongly implied in <a href="http://math.boisestate.edu/~holmes/holmes/head.pdf" rel="nofollow">Holmes</a> (PDF), though not stated outright, that the Axiom of Choice is independent of the other axioms of NFU, though it does state outright on p.95 that the problem of proving there are atoms without choice is open: specifically, if it's adopted, Specker's inconsistency proof becomes a proof that there are atoms.</p> <p>Many set-class theories, in particular NBG and MK, have the axiom of limitation of size, which effectively says that every proper class is the same size. This implies a well-ordering of the class of all sets, and therefore global choice. There is a (very short) <a href="http://en.wikipedia.org/wiki/Axiom_of_limitation_of_size" rel="nofollow">Wikipedia article</a> on this axiom.</p> http://mathoverflow.net/questions/48855/galois-theory-generalization-of-abels-theorem-better-version/49126#49126 Answer by Ian Maxwell for Galois theory: Generalization of Abelâ€™s Theorem? (Better version!) Ian Maxwell 2010-12-12T07:13:33Z 2010-12-12T07:13:33Z <p>A minor note: it suffices to limit oneself to closure under roots of polynomials of form $x^n + b$ or $x^n + x + b$, since any other polynomial of form $x^n + ax + b$ can be transformed into the latter by the change of variables $x = a^{1/(n-1)}y$ (and $a^{1/(n-1)}$ is 'available' by virtue of the former).</p> http://mathoverflow.net/questions/108588/axiomatization-of-the-incidence-geometry-of-the-euclidean-plane Comment by Ian Maxwell Ian Maxwell 2012-10-03T17:18:42Z 2012-10-03T17:18:42Z I just saw that a [very similar question](<a href="http://mathoverflow.net/questions/106156/hilbert-style-axioms-for-euclidean-and-or-hyperbolic-geometry-without-reference-t" rel="nofollow" title="hilbert style axioms for euclidean and or hyperbolic geometry without reference t">mathoverflow.net/questions/106156/&hellip;</a>) was asked recently by David Feldman. In light of this I wouldn't mind if my question were closed. http://mathoverflow.net/questions/108588/axiomatization-of-the-incidence-geometry-of-the-euclidean-plane Comment by Ian Maxwell Ian Maxwell 2012-10-02T03:10:50Z 2012-10-02T03:10:50Z @Ben: Good point! I suppose I could edit the original question to refer specifically to systems taking Playfair as an axiom, if anyone thinks the current version is unclear on that. http://mathoverflow.net/questions/12905/set-theories-that-do-require-the-existence-of-urelements/57259#57259 Comment by Ian Maxwell Ian Maxwell 2011-03-04T20:35:27Z 2011-03-04T20:35:27Z Fair enough; I should more accurately say NFU+Choice. NF itself is inconsistent with choice, which is essentially why NFU was devised. You're right that NF is inconsistent iff NFU can prove the existence of atoms without invoking the axiom of choice. http://mathoverflow.net/questions/14330/decidability-of-the-axiom-of-choice/56432#56432 Comment by Ian Maxwell Ian Maxwell 2011-02-24T01:56:04Z 2011-02-24T01:56:04Z I don't really have a general method in mind. If the original author were alive, I suppose I could just ask him. In the case of V=L I don't know enough about the context in which it was proposed, so I'd have to suspend judgment.