# How to study first order logic, without the use of sets?

I am interested in learning first order logic. In particular, I would like to be comfortable enough with first order logic in order to fully tackle, say, the Axioms of Zermelo-Fraenkel, as in Jech's Set Theory book (I have studied in the past other set theory books, such as Halmos' Naive Set Theory).

The main target is to eventually become comfortable enough with first order logic, and afterwards, with set theory. And then, be comfortable with the basis for the rest of mathematics. Basically, peace of mind.

However, when I try to find the right book (browsing from amazon free pages) for first order logic, I find all books I have tried so far, basically start in the first page using the word "set". The "peace of mind" breaks down then for me, because of course, I want to study first order logic in order to learn set theory later, so I do not want "sets" are a prerequisite for first order logic!

Is there a good book in first order logic that does not use sets at all?

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## migrated from mathoverflow.netJun 22 '14 at 15:34

This question came from our site for professional mathematicians.

There's no avoiding the fact that first order logic requires you to consider various collections (e.g. the collection of symbols). A convenient word for a collection is a "set". If that word bothers you, you can replace the word "set" with the word "collection" throughout your reading, but I don't think you can (or see why you would want to) banish the notion of a collection entirely. –  Steven Landsburg Jun 22 '14 at 14:37
You can use your naive understanding of "set" as a crutch, to support your understanding until you learn what is a "set". This scaffolding system is fairly good. Especially since the use of set theory in the basics constructions of logic (read: interpretation of a language, and so on) are quite naive. Later on you can reinterpret all that you know in a proper context of axiomatic set theory. –  Asaf Karagila Jun 22 '14 at 15:07
(In either way, this question seems to be more fitting to math.SE.) –  Asaf Karagila Jun 22 '14 at 15:08
First order logic is just a formal way of writing down the kinds of statements you will be familiar with in your study of informal mathematics. You do not need to study FOL in any depth in order to use it. Just spend some time googling first order logic and especially translation exercises. I would look to things designed for philosophers. Once you can read and write FOL formulas, open that Jech book and get started. –  mbsq Jun 22 '14 at 15:12
Of course one uses sets. But there is no need of Set Theory. –  André Nicolas Jun 22 '14 at 15:37

Quine's Methods of Logic is a good example of a treatment that makes virtually no reference, even informal, to sets. His treatment is more to informally talk about strings and the truth functional connectives that form new strings; basically, he covers the things that go into the definition of a well-formed formula, but doesn't need to go into talk of the set of wff's.

The trade off here is that while it's easy to talk about proofs like this, the few occasions on which he discusses model theoretic ideas are very sketchy and informal. I never really found his sections on the soundness and completeness of his proof methods informative. So this "set-free" approach is good for learning how to do proofs, but isn't good for particularly deep insights into logic as a formal system; I see this as a feature as much as it's a bug, personally.

Edit: In light of Carl Mummert's comment and answer, I thought I should clarify that I use Quine as an example here; see Carl's comment below for reasons this might not be a great textbook for a mathematics student. While I do consider a treatment of logic in terms of strings and formation rules to be helpful in justifying first order logic's use in a foundational setting, I think Carl's answer also highlights why one can safely be indifferent to the use of mild set-talk.

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While Quine was an exceptional philosopher and mathematician, I am not sure that this sort of book is good for someone newly learning logic. Quine's approach was more philosophical, and his terminology was not standard. Someone who wants to be able to follow modern work would probably be better served by learning from a more modern textbook. –  Carl Mummert Jun 22 '14 at 18:38
@Carl: That's fair. I found the logic I learned from Quine invaluable, personally, when my studies took a decidedly more mathematical bent, but I am aware of this book's shortcomings. My purpose here was not to offer a strong recommendation of Quine's book so much as provide an example of a book that might meet the OP's criteria and highlight what I think that approach doesn't do well. –  Malice Vidrine Jun 22 '14 at 18:56
That is very reasonable. In hindsight I think I voted too quickly; I apologize for needing to add some whitespace to your answer to allow me to vote up. –  Carl Mummert Jun 22 '14 at 19:08
No worries. I am not known for being a skilled communicator, so I assume that if my aim wasn't understood then it's my blunder :) –  Malice Vidrine Jun 22 '14 at 20:53

It is true that most books in first order logic are written using the same sort of mathematical methods as any other area of mathematics, including basic set theory. And why not? Donald Monk's older book Mathematical Logic has a good justification of this practice in the introduction.

For foundational purposes, however, the talk of sets is a sort of "false generality". Let's use ZFC as an example.

• A logic text will talk of a theory as based on a language with a set of symbols. For ZFC the language has only one symbol, $\in$, and a collection of variables. So there is nothing murky about the formulas of ZFC. If you show me a sequence of symbols, I can quickly tell you whether it is a formula of ZFC.

• A logic text will talk about the axioms for a theory being an arbitrary set of formulas. But for ZFC there is a concrete algorithm that enumerates all the axioms. So, rather than being a murky "set", the axioms are very explicitly stated. Now, there are an infinite number of axioms for ZFC - and it is known that no finite set of axioms will suffice. But we could use other set theories, such as NBG, which have a finite set of axioms. That makes the collection of axioms even more explicit.

Most foundational theories are like ZFC in having a completely concrete language and a completely concrete collection of axioms. So the talk of "sets" in logic texts is not needed for these special cases. This phenomenon - that theories of foundational importance tend to be very concrete - is one of the key aspects leading to what is known as finitism in mathematics.

When we turn to the model theoretic aspects of first order logic, however, talk of sets becomes more indispensable. For example, the "completeness theorem" of first order logic shows that every consistent theory in a countable language has a countable model. Thus ZFC has a countable model. There is not any computable countable model of ZFC, however (see here). So, even for very concretely axiomatized theories, there is no way to make the completeness theorem effective. In fact, far more is known about the exact ineffectiveness of the completeness theorem. But, to address that issue, we need to talk about noncomputable sets...

Rather than shoehorning themselves into a "no sets" approach, most books on logic just let the reader who is interested separate the parts of the theory that can be done without any genuine recourse to set theory from the parts that can't.

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