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A lot of textbooks on mathematical logic now rely on set-theoretic tools (models and topology).

  1. do people still care about developing mathematical logic from finitary methods? is there still research in it? are there modern textbooks on this (modern=last 20 years)?

  2. what is the allure of the more modern approach? what would be lost if we could only use finitary/syntactic methods? where can I read more about this?

xwer345
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Speaking as a computer scientist with logic-y interests (who has however been away from academia for about ten years), we certainly care about syntactic methods and properties.

I haven't followed along closely enough to notice the textbook trend you're speaking about -- but if it's there, I wonder if it's not just a case of changed emphasis. There used to be a tradition in mathematically oriented texts to distinguish really carefully and verbosely between when they were doing "finitary" things (proof theory) and when they were assuming standard set theory (model theory) -- e.g., to make sure never to speak of a "set" of formulas while doing proof theory. I can easily imagine that is going away.

It is tempting to point to the failure of Hilbert's program as an explanation for this: nobody these days really look to mathematical logic in the hope of finding a firm foundation that all of mathematics can rest on, and with that motivation gone, the need to keep meticulous track of what is "finitary" and what is not diminishes a lot. The problem with that explanation is that it has been a long time since Hilbert's program died, and it is not obvious why that should start influencing textbooks now in particular.

I think a more immediate cause could be that these days readers who care about being "finitary" (or constructive, etc.) are likely to have first-hand experience with computer programming. This experience, when paired with an interest in mathematics at all, gives a much better and more direct intuition about when something presented as set-theoretic argument can be implemented algorithmically and when it can't, than what the pencil-and-paper mathematicians of yore had. So the textbook doesn't need to be as heavy-handed about explaining what can be mechanized and how, because a reader who cares can be expected to figure out for himself.

The timing works for this explanation. Someone who wrote a textbook 20 years ago would have been unlikely to have grown up with easy access to computers to play with, even if the students they wrote for were by then starting to have computers in their homes. Today's authors of first-edition texts are much more likely to have lived with computers all their life.

hmakholm left over Monica
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    Those of us working in proof theory and foundations do have a very good sense of what things can be done syntactically, and which can't, even if we don't always emphasize it in our writing. – Carl Mummert Jul 19 '16 at 21:00
  • To help broaden my experience: do you recall an example of a text that did what you mentioned - avoided saying 'set of formulas', etc.? In particular, do we need to go all the way back to Kleene to find an example? – Carl Mummert Jul 19 '16 at 21:04
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    @Carl: Oh, I'm not suggesting that computer scientists are the only ones. Just speculating that the wide availability of computers may have made textbook authors more willing to trust that interested students already have such a sense too, so they don't need to emphasize it as much. – hmakholm left over Monica Jul 19 '16 at 21:09
  • @Carl 2: I haven't seen one personally. I remember Mendelson apologizing in his preface for not following such a style. But that is, of course, half a century ago. (In general, I'll freely admit I'm reaching rather wildly in an attempt to imagine a sense in which the OP's premise might be true). – hmakholm left over Monica Jul 19 '16 at 21:14
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    Monk's logic book from the 1970s also has an explanation in the introduction about how the study of "mathematical logic" means using general mathematical methods to study logic. But to find a mathematical logic book that actually does limit itself to finitistic methods for logic, the most recent one I know is Kleene's book from 1950. Maybe someone else knows something more recent. – Carl Mummert Jul 20 '16 at 00:31