If logic/type theory/model theory don't depend on set theory

Question

I am at the beginning of trying to understand how to read foundations, but one thing that keeps tripping me up is that, when authors introduce theories that seem like they should proceed or replace set theory, a lot of times those authors will refer to "the set of symbols" or "the set of relations" or some other set that just describes a relevant collection of things the theory has defined. Why is this allowed? Do we actually need set theory to understand any foundational theory, or are there theories that are true, independant alternatives?

Answer 1

There are (at least) two rather different subjects often labeled "logic". The first is the analysis, by standard mathematical methods, of deductive (mathematical) reasoning. The second is providing a foundation on which all of mathematics is to be built.

When doing logic in the first sense, we can reasonably use whatever mathematical ideas and methods are appropriate, just as in any other branch of mathematics. The fact that we're analyzing reasoning rather than, say, planetary motion or financial markets or whatever, need not hobble our choice of mathematical techniques. So, when I teach logic (in this first sense), I use sets, mathematical induction, Zorn's lemma, etc., just as in any other mathematics course.

When doing logic in the second sense, we should not use methods beyond what our foundation (under construction) provides. So there should be some axioms and rules of inference and probably definitions or abbreviations, and we should proceed using these. We'll still need some basic, undefined and unaxiomatized concepts, like substituting a name for a variable in a formula (although even that has been axiomatized in some foundational systems). But these primitive notions, preceding even the axioms, should be extremely simple ones, essentially just manipulating formulas in trivial ways.

You wrote that you want to read foundations, which sounds like the second meaning of "logic", but your description of the books you're studying sounds more like the first meaning. Needless to say, if the author is trying to do one thing (logic in the first sense) and you're looking for something else (logic in the second sense), you'll be disappointed.