What do we call a set that has one or fewer elements? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-22T19:34:52Z http://mathoverflow.net/feeds/question/107665 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107665/what-do-we-call-a-set-that-has-one-or-fewer-elements What do we call a set that has one or fewer elements? user1837 2012-09-20T09:28:39Z 2012-09-20T23:32:27Z <p>Suppose a set $X$ has the property that for all $a,b \in X$, we have that $a=b$. Such a set is either empty, or it's a singleton set. In other words, it has one or fewer elements. What do we call this?</p> http://mathoverflow.net/questions/107665/what-do-we-call-a-set-that-has-one-or-fewer-elements/107669#107669 Answer by James Cranch for What do we call a set that has one or fewer elements? James Cranch 2012-09-20T10:24:00Z 2012-09-20T10:24:00Z <p>The language of homotopy type theory suggests <a href="http://ncatlab.org/nlab/show/%28-1%29-groupoid" rel="nofollow">names</a> for a set with the "any two objects are equal" property. You can call it a <em>$(-1)$-groupoid</em> or a <em>$(-1)$-type</em> or a <em>$h$-proposition</em>.</p> <p>It may sound like the names involving $-1$ suffer from the same weaknesses as Asaf's suggestion <em>set of size at most one</em>, namely that it depends on numbers. But in fact there is an inductive definition of $n$-type (starting with $-2$), and it is easy to unpack it to get this definition.</p> <p>Unfortunately, to many audiences these names are currently likely to cause more confusion than joy. Still, maybe someone should try to popularise them outside their current domains.</p> http://mathoverflow.net/questions/107665/what-do-we-call-a-set-that-has-one-or-fewer-elements/107671#107671 Answer by Tom Leinster for What do we call a set that has one or fewer elements? Tom Leinster 2012-09-20T10:50:52Z 2012-09-20T14:37:02Z <p>In category theory, subobjects of the terminal object are sometimes called <strong>subterminal</strong>. An equivalent definition: an object $A$ is subterminal if and only if for all objects $B$, there is at most one map $B \to A$. (Note that in the case where the category is $\mathbf{Set}$ and $B = 1$, this says exactly that $A$ has at most one element.) The sets you mention are the subterminal sets.</p> <p>(But still, if I was writing a paper in which I only mentioned such sets a handful of times, I'd probably just say "sets with at most one element".) </p> http://mathoverflow.net/questions/107665/what-do-we-call-a-set-that-has-one-or-fewer-elements/107683#107683 Answer by Andreas Blass for What do we call a set that has one or fewer elements? Andreas Blass 2012-09-20T13:04:16Z 2012-09-20T13:04:16Z <p>I've occasionally used "subsingleton". I think this is OK in the context of classical logic, which is what you're using since you say that such a set must be a singleton or empty. In the context of constructive logic, "subsingleton" doesn't look so good, since it suggests that the set is a subset of some singleton, which need not be the case. Unfortunately, I don't know a better term that avoids constructively unwanted suggestions.</p> http://mathoverflow.net/questions/107665/what-do-we-call-a-set-that-has-one-or-fewer-elements/107696#107696 Answer by Gerhard Paseman for What do we call a set that has one or fewer elements? Gerhard Paseman 2012-09-20T15:57:28Z 2012-09-20T15:57:28Z <p>I would explain the concept and then name it. Since the idea is to capture the uniqueness of any potential member of the set X, I would suggest describing such X as "singular". Be sure not to use "singular" for anything else in that context.</p> <p>Gerhard "Caution: Alternate Words At Work" Paseman, 2012.09.20</p>