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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?

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It's not exactly your question, but I sometimes refer to an object as being unique, even when I don't know that it exists. (Exists <=> lower bound of 1, unique <=> upper bound of 1.) It's annoying that this is nonstandard enough that I always have to follow up "Note: I'm not saying that it exists!" –  Allen Knutson Sep 20 '12 at 10:10
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To me, "singleton or empty" or "a set with at most one element" look just fine; but if you are unhappy with this for any reason, there are other options of this sort: say, "a subset of a singleton". You can even create a designated name for it, like "sub-singleton set". –  Seva Sep 20 '12 at 10:41
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For colorful terminology, you could go with coupleless, non-reproductive, lonely, tango-incapable (with the generalization to crowdless for sets with < 3 elements). In all honesty, I'd probably just use set with at most one element. –  Michael Joyce Sep 20 '12 at 14:36
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one could also use "doubleton-free". ("doubleton-free set" is actually just the trivial first instance of a more interesting sequence, the next two elements of which are "triangle-free graph" and "tetrahedron-free 3-uniform hypergraph".) –  Terry Tao Sep 20 '12 at 16:03
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This really has nothing to do with set theory or logic, so I removed the tags. –  Andres Caicedo Sep 20 '12 at 18:33
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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.

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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.

Gerhard "Caution: Alternate Words At Work" Paseman, 2012.09.20

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In category theory, subobjects of the terminal object are sometimes called subterminal. 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.

(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".)

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The language of homotopy type theory suggests names for a set with the "any two objects are equal" property. You can call it a $(-1)$-groupoid or a $(-1)$-type or a $h$-proposition.

It may sound like the names involving $-1$ suffer from the same weaknesses as Asaf's suggestion set of size at most one, 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.

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.

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This sounds a bit like using a bazooka to kill a mosquito. :-) –  Qfwfq Sep 20 '12 at 11:44
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That's a theological matter, Qfwfq. If you think that mathematics is built out of sets, it is. If you think it's built out of types, it isn't. –  James Cranch Sep 20 '12 at 12:11
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