Most books on introductory logic seem to work on a metatheory where infinite sets are allowed to exist. This seems unnecessary: everything humans do is finite, so seems like it should be enough to assume existence of only finite things. The more we accept in our (meta)theory the more likely it is that something goes wrong.

However, in the book *Lectures in Logic and Set Theory. Volume 1* by Tourlakis, the author says:

If it were not for Gödel’s incompleteness results, this position – that meta mathematical techniques must be finitary – might have prevailed. However, Gödel proved it to be futile, and most mathematicians have learnt to feel comfortable with infinitary metamathematical techniques, or at least with $\mathbb{N}$ and induction. Of course, it would be reckless to use as metamathematical tools “mathematics” of suspect consistency (e.g., the full naïve theory of sets).

It seems to me that Gödel's results don't make the finitary position any more futile than the infinitary ones. We can't know consistency of strong enough systems anyway, so the finitary position still seems safer, even if not as convenient. It's also strange that it seems to be very hard to find recent textbooks who work with finitary metatheory. (unless i'm searching wrongly?). Are there other disadvantages to a finitary metatheory or am i missing something?