I have been reading introductory texts on first order logic (for example, Leary&Kristiansen). All of them used concepts that I have heard in set theory courses – ordered pairs, functions, bijections, isomorphism and so on.
I have read a lot of material in Math.StackExchange on set theory and first order logic and their interplay. I understand that we cannot define mathematics from nothing – we have to have primitive concepts. My problem is with understanding which are primitive concepts and which are not. Of course, different books might take these to be different, but still – maybe there are commonly accepted principles and notions that noone doubts.
For example, I am willing to accept that strings exist, that they can be glued together or separated, also I am willing to accept recursion and induction. I am also willing to accept counting numbers (which could as well be infinite: I, II, III, …).
Question 1: As far as I have read and understood – sets in first order logic are different than those in set theory. But how so? At first I thought that it is because sets in first order logic are finite by definition and are basically just collections of finite terms, strings, and so on. Then, paradoxes that arise in set theory due to infinities do not arise in logic. But on the other hand, we use counting numbers and then, for example, the number of terms can be infinite.
Question 2: Are sets, ordered pairs, functions, bijections – primitive notions (by primitive notion I understand concept that is not defined) in first order logic (at least in the one that most of the mathematicians use)?
Question 3: If sets, ordered pairs, functions are indeed primitive notions then are they different from set theoretical definitions? If yes, then in which way? If no, then why define these concepts once again in set theory if we had them in the language of set theory anyway?
Question 4: If sets, ordered pairs, functions are not primitive notions in first order logic then how are they defined?
I would appreciate any comments and discussions about this topic.
Best Answer
The questions assumes that there is some notion of "set" in first-order logic itself, but there is not. We use sets to study first-order logic, particularly the semantics (models) aspect. But these are part of the metatheory we use to study logic, not really part of "first order logic". For example, if we look at the first-order theory of groups, there is nothing in it about "sets".
If we look more at the syntactic (proofs) side, we can get by with a much weaker metatheory, one which only needs to manipulate strings. Theories often used for this purpose include Peano arithmetic and the weaker Primitive Recursive Arithmetic. In these theories, there aren't directly any "sets", just natural numbers, although these theories have ways to talk about functions from numbers to numbers and, as such, indirectly talk about some kinds of sets.
The really fundamental concepts in first-order logic are alphabet, signature, language, theory, formal proofs/derivability, and models/satisfiability. All but the last of these can be very satisfactorily studied using Peano arithmetic as our metatheory. Once we move to studying models - which are again a fundamental part of first-order logic - we usually find it more satisfactory to work in a stronger metatheory that is able to construct and work with models more directly.
On the nature of logic
The other thing about this particular question: it is common for people first studying mathematical logic to think that the main purpose of studying logic is to find the most primitive objects of mathematics and then to rebuild mathematics from these primitive objects -- this is the foundational aspect of logic.
That is indeed one aspect of mathematical logic, but not the only one by far. Historically, the foundational aspect was of particular interest around the turn of the 20th century, but it is not of such primary interest any longer. From the contemporary viewpoint, another purpose of mathematical logic is simply to understand mathematics better by using techniques that have come to be called "mathematical logic". I think that, for historical reasons and because it's interesting, the foundational aspect tends to be slightly over-emphasized in introductory materials.
For example, another common and important thread in mathematical logic is definability - the study of which aspects of mathematical structures can be expressed in which formal languages. This thread runs very heavily through computability theory and model theory, and is also found in set theory and proof theory.
Yet another common thread is an interest in the mathematical objects of logic for their own sake: some logicians study sets because they like sets, not as a way to study foundations. Some study computability because they like computability, without much interest in philosophical aspects. Some research topics in model theory are essentially indistinguishable from abstract algebra or analysis.
The foundational aspect of logic is still important, of course, and there are still people who work primarily on foundations. But the idea that mathematical logic will provide some sort of rock-solid foundation to all the rest of mathematics is not really part of the contemporary study of foundations. Instead we think about a range of theories, each suitable for its own foundational purpose. For studying the semantics of first-order logic, we need a theory that includes some way to handle models, which are particular kinds of sets.
As the shift from a mainly foundational viewpoint to a more broadly mathematical viewpoint occurred, several mathematical logic books from the mid 20th century included detailed explanations in the introduction about why they use advanced mathematical methods to study logic. One good treatment of this topic is in Monk's logic book, which can be found pretty cheaply these days.
The purpose of this section, which may be a slight digression, is to explain that one reason that it is not easy to see how logic is developed "out of nothing" from absolutely first principles is that, often, that isn't the goal that contemporary logicians have in discussing logic. They aren't necessarily trying to develop logic and mathematics from absolutely first principles.