Why does a scientist care why his theory doesn't predict the existence of God?
Limitative results are just as important, if not more important, science usually can show theories wrong, but it cannot show them right - at best they are just confirmed by experiment. If you can show that certain things are impossible, that is progress forward.
Yep! You observe "evidence for the unbounded view", but fail to present such evidence to the humble goyim.
Depends on which unbounded view you mean - there are quite a few. In cosmology you have the concept of "eternal inflation". In some multiverse theories, you almost always end up with some unbounded parameters. In multiverse theories which even vary the laws themselves, you again end up with the infinity of structures or possible computations - this infinity is almost always at the meta-level, not at the local level which you experience, however this is important, if the "everything" was limited in entities, it would be a (mathematical) miracle (highly improbable event) if anything like you or me would ever have any chance of existing at all. In the computational theory of mind, one yet again ends up with this globally unbounded, but locally finite ontology which predicts that the shape of physical law will always have certain properties (mostly confirmed by quantum mechanics so far). Terms to google for each case:
1) "eternal inflation" (
http://arxiv.org/abs/astro-ph/0101507 )
2) "Ultimate Ensemble" (
http://arxiv.org/abs/0704.0646 http://arxiv.org/abs/1008.1066 http://arxiv.org/abs/gr-qc/9704009 ), "Algorithmic theories of everything" (
http://arxiv.org/abs/quant-ph/0011122 ), "Universal Dovetailer Argument"/"Movie Graph Argument" (
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html http://iridia.ulb.ac.be/~marchal/publications/CC&Q.pdf ) or if you want an easier to grasp understanding of this concept, read the novel "Permutation City", you might even like it as it takes a partially ultrafinitist view
near the end.
Note that I intentionally avoided the cases where scientists put real numbers in their physical theories - you would reject such theories out of hand, however the problem is that in almost all cases where you take reality seriously, you end up with at least the countable infinity - this is not easily avoidable, even if it can be locally avoided, just never globally.
However, don't dispair - if you do show arithmetic inconsistent, everyone will be forced to completly rethink their theories, but for now, almost no-one doubts that arithmetic or computation leads to inconsistencies. If I ever saw an inconsistency proof of arithmetic, I would be shocked and it would shake my world-view, but then I would read it and understand it, and if it was correct, I would accept it and continue to move on from there as now our knowledge is more complete and we can form better theories. However, for now, I'll continue my
religious bet in the consistency of arithmetic (the same as almost any scientist or mathematician) and reason from there.