Definition talk:Initial Segment

Definition does not need to require that $\prec$ form a poset. Initial segments are very important to foundational relations and the axiom of regularity. See Axiom of Foundation (Strong Form). --Andrew Salmon (talk) 07:13, 22 August 2012 (UTC)


 * We've had this conversation before.
 * In the context from which your source work approaches it, maybe not. In the context from which my source work approaches it, it does.
 * If you need to add the definition of initial segment that you require in order to progress your own thread, then add the definition as an "also defined as", but I would contend that the object you are defining is a different object which happens to have the same name. --prime mover (talk) 09:03, 22 August 2012 (UTC)


 * IMO adding stuff under 'Also defined as' is not the way to go when it constitutes a generalisation, or a properly different approach. In such cases, a page with multiple definitions needs to be crafted, or wholly different pages altogether, linked via a disambiguation. I vote for the latter option in this situation (and maybe also for the other page that recently got such a 'A d a' section - I forgot the name). Of course, the two are interlinked in that one is an instance of the other, but such happens all the time. --Lord_Farin (talk) 09:16, 22 August 2012 (UTC)


 * Let me add to that the suggestion that the poset def be transcluded onto the general page, like on Definition:Small Class, or a construct like Definition:Open Set. --Lord_Farin (talk) 09:20, 22 August 2012 (UTC)


 * Yes that would probably work. As long as we don't lose the fact that the "conventional understanding" is that $\prec$ is an ordering but that the more modern abstract approach lifts this requirement and merely stipulates that it be foundational. That the traditional $\preceq$ is a foundational relation with transitivity union with the diagonal relation can be deduced from there. --prime mover (talk) 13:53, 22 August 2012 (UTC)


 * It need not be foundational. $<$ for Real numbers is not foundational.  It can just be an abstract definition for any relation $\mathcal R \subseteq A \times A$. --Andrew Salmon (talk) 17:44, 22 August 2012 (UTC)


 * So what constraints are necessary on this relation for this to be valid? --prime mover (talk) 18:59, 22 August 2012 (UTC)


 * Actually, nothing is necessary. --Andrew Salmon (talk) 23:40, 22 August 2012 (UTC)