Terminology for a property that holds in the finite but not infinite case?

Question

(I apologize if this is a duplicate, but I don't know what terms to search for. Please feel free to close this if this has already been asked.)

There are some properties of finite objects that don't scale up to the infinite case. For example, any finite set of real numbers must have a least element, though an infinite set of real numbers needn't have a least element. Similarly, any meet semilattice of finite height is also a join semilattice, but when extended to the infinite case this no longer holds true.

Is there are a term for properties like these that hold in the finite case but not the infinite case?

Thanks!

Answer 1

Not in general, no: such properties don’t form a natural class, so there’s no good reason to have a general term for them. In your first example, for instance, it isn’t that the property non-empty sets have least elements doesn’t scale up: it’s simply that $\Bbb R$ is not well-ordered by the usual order. In $\omega_1$, say, that property does scale up.

I don’t think that you’re really looking at a type of property at all, but rather at a class of theorems of the form

if X is a finite so-and-so, then thus-and-such is true of X,

where in many cases thus-and-such may be true of an infinite so-and-so $-$ it just isn’t guaranteed.