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In explanations occurring in proofs such as, e.g., "This way we reduce unboundedly many objects of type ... to constantly many ones.", is "unboundedly many" a correct term? Of would you write "This way we reduce boundlessly many objects of type ... to constantly many ones."?

Another example is found in https://arxiv.org/abs/1812.00183 :

A Scheme to Verify Services with Unboundedly many Clients using NuSMV. We study model checking of client-server systems, where the servers offer several types of services that may depend, at any time, on how many clients of specific types are active at that time. Since there are unboundedly many clients, the state space of such systems is infinite, rendering specification and verification hard. ...

Yet another example from http://books.google.de/books?id=viAzDwAAQBAJ :

There is a corresponding set of boundlessly many pairs of numbers: (1, 10), (10, 100), and so on. We cannot list all these pairs.

Of course, google has its counts in favor of one variant, but lots of folks writing the two variants are not native English speakers.

PS. An example for "constantly many" from http://arxiv.org/abs/1208.5639 :

Convex Integer Optimization by Constantly Many Linear Counterparts.

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In contemporary mathematical English, the two choices would be "infinitely many" and "finitely many". Possibly "unboundedly many" could work, but it would be odd, and would invite confusion. "Boundedly many" invites far worse confusion, since it might be that there are infinitely many, but "bounded" by some measure (e.g., size).

EDIT: and, to talk about a family of cases, perhaps parametrized, e.g., by time (as in the case of computer memory), one keyword would be uniformly bounded, or uniformly finite.

paul garrett
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