(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?


  • $\begingroup$ If you definitely know that the property doesn't hold in the infinite case, you can say something like "at most finitely". $\endgroup$ – Dave L. Renfro Mar 9 '12 at 21:35
  • $\begingroup$ I would say a "finite property" for this. $\endgroup$ – Patrick Da Silva Mar 9 '12 at 21:58
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    $\begingroup$ @Patrick: I wouldn’t. Would you also say that a property common to many green things is a green property? $\endgroup$ – Brian M. Scott Mar 10 '12 at 7:53
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    $\begingroup$ @Brian : No, but i would say it for finite things. :) $\endgroup$ – Patrick Da Silva Mar 11 '12 at 2:20
  • $\begingroup$ I asked a related question, which takes the "any meet semilattice of finite height is also a join semilattice" part to explain the motivation. However, my question also remarks that this statement at least required further clarification, because it is not true in the literal sense. $\endgroup$ – Thomas Klimpel Oct 19 '12 at 14:31

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.

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  • $\begingroup$ That would be the most sensible way of thinking about it, but since OP was looking for a name for it, I guess I just gave him one. +1 though, this is what I would've said if I didn't feel like creating names. $\endgroup$ – Patrick Da Silva Mar 12 '12 at 15:13

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