Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adv. In a finite manner.
from the GNU version of the Collaborative International Dictionary of English
 adv. In a finite manner or degree.
from The Century Dictionary and Cyclopedia
 In a finite manner or degree; within limits; to a certain degree only.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 adv. with a finite limit
Etymologies
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Examples

A theory T is defined to be decidable if there is an effective procedure of deciding whether any given sentence s belongs to T, (where an 'effective procedure' is generally defined to be a finitelyspecifiable sequence of algorithmic steps).

Similarly, there are economic theories predicting that firms will set price equal to average cost, that individuals will discount future payments at the rate of interest, and that no one will cooperate in a finitelyrepeated Prisoners 'Dilemma game.
The Marginal Myth, Bryan Caplan  EconLog  Library of Economics and Liberty

In any finite region of space, matter can only arrange itself in a finite number of configurations, just as a deck of cards can be arranged in only finitely many different orders.

It feels a lot like a finitely repeated prisoner's dilemma: if everyone's going to fink in the end, then we can't do any better at the beginning but to fink.

But a key fact is that the roots are always countable — and, more importantly, there can only be finitely many roots in a given closed and bounded region.
Wolfram Blog : Mathematica 7, Johannes Kepler, and Transcendental Roots

"3D works best in a very claustrophobic environment where you really feel finitely that the walls are only a few feet away," Cameron explains.
Cameron on Producing His Underwater 3D Drama 'Sanctum' « FirstShowing.net

An atheist could still try to deny the existence of the idea of God on the grounds that there are only finitely many natural laws, i.e. ideas, in the platonic world, but this can be refuted by using the Goedel theorem: A theory of everything should explain mathematics, and by Goedel´s theorem there is no closed mathematical system which contains the arithmetics.

They make mistakes, and they're finitely human instead of being superheroes.
Holly Robinson: Our TV Sleuths Need to Take a Page from British Mysteries

We observe our universe to be such, if current cosmological theory is right, as to have had a beginning finitely long ago, followed by an era in which no part of it could have sustained life.

You know from your extensive reading of Goedel, far more extensive than mine, that any finitely specified formal consistent system large enough to contain arithmetic cannot justify its axioms within that system.
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