candidate for rigorous _mathematical_ definition of "canonical"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:01:37Z http://mathoverflow.net/feeds/question/20154 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20154/candidate-for-rigorous-mathematical-definition-of-canonical candidate for rigorous _mathematical_ definition of "canonical"? Kevin Buzzard 2010-04-02T10:00:57Z 2012-01-02T22:24:43Z <p>In this question: <a href="http://mathoverflow.net/questions/19644/what-is-the-definition-of-canonical" rel="nofollow">http://mathoverflow.net/questions/19644/what-is-the-definition-of-canonical</a> , people gave interesting "philosophical" takes on what the word "canonical" means. Moreover I percieved an underlying opinion that there was no formal mathematical definition. </p> <p>Whilst looking for something else entirely, I just ran into Bill Messing's post</p> <p><a href="http://www.cs.nyu.edu/pipermail/fom/2007-December/012359.html" rel="nofollow">http://www.cs.nyu.edu/pipermail/fom/2007-December/012359.html</a></p> <p>on the FOM (Foundations of Mathematics) mailing list. I'll just quote the last paragraph:</p> <p>"It is my impression that there is very little FOM discussion of either Hilbert's epsilon symbol or of Bourbaki formulation of set theory. In particular the chapitre IV Structures of Bourbaki. For reasons, altogether mysterious to me, the second edition (1970) of this book supressed the appendix of the first edition (1958). This appendix gave what is, as far as I know, the only rigorous mathematical discussion of the definition of the word "canonical". Given the fact that Chevalley was, early in his career, a close friend of Herbrand and also very interested in logic, I have guessed that it was Chevalley who was the author of this appendix. But I have never asked any of the current or past members of Bourbaki whom I know whether this is correct."</p> <p>It's a 4-day weekend here in the UK and I'm very unlikely to get to a library to find out what this suppressed appendix says. Wouldn't surprise me if someone could find this appendix on the web somewhere though! Is there really a <em>mathematical</em> definition of "canonical"??</p> <p>NOTE: if anyone has more "philosophical" definitions of the word, they can put them in the other thread. I am hoping for something different here.</p> http://mathoverflow.net/questions/20154/candidate-for-rigorous-mathematical-definition-of-canonical/20156#20156 Answer by Thomas Sauvaget for candidate for rigorous _mathematical_ definition of "canonical"? Thomas Sauvaget 2010-04-02T10:22:35Z 2010-04-02T10:22:35Z <p>There are scanned notes in french that were used for the initial text of <em>Théorie des Ensembles</em> <a href="http://mathdoc.emath.fr/archives-bourbaki/feuilleter.php?chap=2_REDAC_E1" rel="nofollow">on the Bourkaki Archives website</a>. </p> <p>In particular there are indeed notes by Chevalley named <em>Livre I. Théorie des ensembles Chap. IV (état 7 ?) Structures (53 p.)</em> which seem at first glance to define "canonique" in the broader context of "transport de structures, idendifications" (see exemple 1 at the bottom of page 19 of that file). </p> http://mathoverflow.net/questions/20154/candidate-for-rigorous-mathematical-definition-of-canonical/20159#20159 Answer by François G. Dorais for candidate for rigorous _mathematical_ definition of "canonical"? François G. Dorais 2010-04-02T12:19:49Z 2010-04-02T14:51:06Z <p>Although the Bourbaki formulation of set theory is very seldom used in foundations, the existence of a definable Hilbert $\varepsilon$ operator has been well studied by set theorists but under a different name. The hypothesis that there is a definable well-ordering of the universe of sets is denoted V = OD (or V = HOD); this hypothesis is equivalent to the existence of a definable Hilbert $\varepsilon$ operator.</p> <p>More precisely, an <a href="http://en.wikipedia.org/wiki/Ordinal_definable_set" rel="nofollow">ordinal definable set</a> is a set $x$ which is the unique solution to a formula $\phi(x,\alpha)$ where $\alpha$ is an ordinal parameter. Using the reflection principle and syntactic tricks, one can show that there is a single formula $\theta(x,\alpha)$ such that for every ordinal $\alpha$ there is a unique $x$ satisfying $\theta(x,\alpha)$ and every ordinal definable set is the unique solution of $\theta(x,\alpha)$ for some ordinal $\alpha$. Therefore, the (proper class) function $T$ defined by $T(\alpha) = x$ iff $\theta(x,\alpha)$ enumerates all ordinal definable sets.</p> <p>The axiom V = OD is the sentence $\forall x \exists \alpha \theta(x,\alpha)$. If this statement is true, then given any formula $\phi(x,y,z,\ldots)$, one can define a Hilbert $\varepsilon$ operator $\varepsilon x \phi(x,y,z,\ldots)$ to be $T(\alpha)$ where $\alpha$ is the first ordinal $\alpha$ such that $\phi(T(\alpha),y,z,\ldots)$ (when there is one).</p> <p>The statement V = OD is independent of ZFC. It implies the axiom of choice, but the axiom of choice does not imply V = OD; V = OD is implied by the axiom of constructibility V = L.</p> <hr> <p>When I wrote the above (which is actually a reply to Messing) I was expecting that Bourbaki would define canonical in terms of their $\tau$ operator (Bourbaki's $\varepsilon$ operator). However, I was happily surprised when reading the 'état 9' that Thomas Sauvaget found, they make the correct observation that $\varepsilon$ operators do not generally give canonical objects.</p> <p>A term is said to be 'canonically associated' to structures of a given species if (1) it makes no mention of objects other than 'constants' associated to such structures and (2) it is invariant under transport of structure. Thus, in the species of two element fields the terms 0 and 1 are canonically associated to the field F, but $\varepsilon x(x \in F)$ is not since there is no reason to believe that it is invariant under transport of structures. They also remark that $\varepsilon x(x \in F)$ is actually invariant under automorphisms, so the weaker requirement of invariance under automorphisms does not suffice for being canonical.</p> <hr> <p>To translate 'canonically associated' in modern terms:</p> <p>1) This condition amounts to saying that the 'term' is definable without parameters, without any choices involved. (Note that the language is not necessarily first-order.)</p> <p>2) This amounts to 'functoriality' (in the loose sense) of the term over the <a href="http://ncatlab.org/nlab/show/core" rel="nofollow">core groupoid</a> of the concrete category associated to the given species of structures.</p> <p>So this seems to capture most of the points brought up in the answers to the <a href="http://mathoverflow.net/questions/19644/what-is-the-definition-of-canonical" rel="nofollow">earlier question</a>.</p>