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Nowadays, mathematicians have a number of ways of dealing with infinities, prominently the notions of cardinalities and ordinal numbers. As such, I wondered whether we would incorporate such notions into our natlangs.

When listing cardinalities by order, the most remarkable "leap" is between finite cardinalities and infinite cardinalities. As such, I'd consider it natural to treat finite cardinalities as "paucal" and to treat infinite cardinalities as "plural".

Of course, our current English distinguishes only zero/one/two-or-more, but would any natural language gain a reason to declare infinities as a separate grammatical number? If so, what would such reason be?

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I wouldn't expect that in a natlang, no.

When a language has a mandatory marking that goes on every noun (or verb or etc), it's always something that's frequently important or useful to distinguish. It might be important semantically (number marking), or syntactically (case marking), or it might add useful redundancy (gender marking), but it always serves some kind of purpose. If it didn't, people wouldn't go to the extra effort!

And infinities just don't come up very often in day-to-day life. It's not very often that we have to distinguish between "finitely many books" and "infinitely many books", and when we do, "infinite" or "infinitely many" is an easy enough modifier to use.

If you wanted a marking like that to evolve naturalistically, you'd need people to be making a distinction between "finitely many cups of coffee" and "infinitely many cups of coffee" often enough that it's worth incorporating that distinction into the noun itself.

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I think this question is posed from the wrong perspective.

It is of course perfectly possible to describe a system of grammatical number that distinguishes between finite and infinite cardinalities, but describing those grammatical numbers as paucal and plural respectively would be perverse given the definition of paucal as "referring to a few of something".

Instead such a number system would be better described as having two grammatical numbers: finite and infinite (each of which could be further subdivided if you so chose, e.g. with finite having singular, paucal, plural subnumbers, and infinite having countable and uncountable subnumbers etc).

As Draconis notes, this system would not be naturalistic, as infinities do not occur in every day life and, to the extent they do, are not distinguished from sufficiently large finite quantities (obviously the point at which the quantity becomes sufficiently large depends on what is being measured). The only reason this would evolve in a natural language is if people are regularly dealing with genuine infinities and distinguishing them from even arbitrarily large finite numbers (perhaps the culture ascribes set theorists an almost religious importance).

Of course, not all conlangs need be naturalistic, and in a philosophical language inclined towards certain kinds of mathematics for instance this could certainly be an interesting feature to include despite the lack of naturalism.

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