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For example, in computer science, there can be zero, one, two, etc. parameters to a computer program, and this is called its "arity". Sets can be countable or uncountable. Is there some word I can use to say "this set's ___" is infinite/finite, or "this set has an infinite __". For example, although this sounds terrible, "this set's finitude is infinite" or "this set has an infinite finitude".

Anonymous
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    I think "finitude" is the right word; I just wouldn't use it like that (in both of your examples, you might just as well say "this set is infinite."). But if $f$ was a function which took on different values on finite sets than on infinite ones, I wouldn't have a problem saying "the finitude of a set is determined by its image under $f$" or some such. – Micah Aug 06 '13 at 18:24
  • Yeah, I'm trying to find a property name that makes sense for a computer program I'm writing. I don't want to just pick one or the other and say "discrete: true", or "continuous: false", or define both and set them as opposites... seems messy. I'd rather just use a single property name that describes the collection. – Anonymous Aug 06 '13 at 18:25
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    Accepting answers less than 1 hour after the question was asked has some detrimental effects. In the present case, note that I never ever saw the word "population" used in the setting which interests you. Commonly used are the expressions "The size of this set is finite/infinite" and (the simpler the better) "This set is in/finite". – Did Aug 06 '13 at 20:36
  • I'm happy to change the accepted answer if you have a better one. Size doesn't seem better. – Anonymous Aug 06 '13 at 20:46

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The "size" of a set is known as its cardinality. This may be a more fine-grained term than you are looking for, however - it distinguishes between finite sets of different sizes, and there are a hierarchy of infinite cardinals.

Anthony Carapetis
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The word population seems a natural choice, as in "This set's population is finite/infinite."

Membership works as well.

gamma
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"Power" would have worked a long time ago. If you don't like "size" or "cardinality", how about "magnitude"?

Rob Arthan
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  • Its not about liking or not liking "cardinality." The problem the OP is identifying is that "cardinality" is sometimes too fine grained. – goblin GONE Apr 25 '14 at 10:52
  • No that is not what the OP has identified. If it was, I would simply have recommended "X is finite" or "X is infinite". However, the OP wants to assign a property to a set X and to qualify that property as finite or infinite, so something like "cardinality" or "size" or "magnitude" is called for. So it is about or not liking standard mathematical terminology like "set of finite size" or "set of finite cardinality". – Rob Arthan Apr 27 '14 at 00:56