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I am taking a 'mathematical logic' course this semester, but the material we are given seems a bit superficial as it goes directly to the methods to simplify formulas. What I'm looking for is a material that talks about the construction of logic (from basic set theory maybe?) that is a little bit more rigorous than our course and that starts from zero. I have been searching on the internet, but I can't seem to find anything interesting. It'd be nice if anyone could give me some links (in French or English) about that.
Thanks in advance.

Watson
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truthseeker
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  • Related : http://math.stackexchange.com/questions/264913/book-on-the-rigorous-foundations-of-mathematics-logic-and-set-theory?lq=1, http://math.stackexchange.com/questions/4170/good-books-on-mathematical-logic?rq=1 – Watson Mar 05 '16 at 10:00
  • You don't construct mathematical logic by set theory. You construct it using metamathematical objects (for instance, metamathematical natural numbers). Actually, you construct set theory using mathematical logic (first-order logic). There are some circular treatments though, but they're circular, so... – user40276 Mar 05 '16 at 10:01
  • If you want a good book (with some philosophical problems as any other books in logic) in classical logic take a look at Shoenfield's book – user40276 Mar 05 '16 at 10:03
  • Stephen Simpson, Mathematical Logic (2013). – Mauro ALLEGRANZA Mar 05 '16 at 10:11
  • @user40276: Exactly. See http://math.stackexchange.com/a/1334753/21820. In fact, the bare minimum that we need, to be able to even talk about any reasonable kind of logic and proofs, is PA or equivalent in the meta-system. And no one can ever define any sort of equality or conditional without already relying on our prior knowledge of equality or conditional! – user21820 Mar 05 '16 at 12:04
  • Thank you everyone for your answers. – truthseeker Mar 05 '16 at 15:49
  • I would choose Van Dalen's "Logic and Structure" as a rigorous introduction. – Ak9 Mar 05 '16 at 21:15
  • @user21820 I don't understand what you mean by "And no one can ever define any sort of equality or conditional without already relying on our prior knowledge of equality or conditional!" nor what you mean by "to define". If you accept a metatheory that is based on spoken language or evident concepts (of course, this is subjective), intensional equality is defined as well as sense and reference (as in Frege terminology). A further deduction "calculus" can be implemented by epistemic logic (philosophy) too. – user40276 Mar 06 '16 at 17:43
  • @user40276: That's exactly what I'm saying. You have to a priori accept a metatheory that already has the concepts of equality and conditionals. In other words they cannot be reduced or ontologically justified in any way that is not circular. And so if you already accept equality and conditionals in a meta-theory, there is no reason to ever attempt to justify them. A lot of people make this mistake when they claim that you can define the conditional say using truth tables. Also, proofs require the notion of (finite) strings, which again you must have in the meta-theory already. – user21820 Mar 06 '16 at 18:38
  • @user21820 Oh!Yes, I agree and understand what you mean. Actually, the truth table just define the reference of the conditional, so unless you're in a kind of extensional metatheory, this does not define it entirely. – user40276 Mar 06 '16 at 20:45
  • @user40276: And this line of thinking tells us that the bottom line is that no formal system in general can be justified ontologically except by a sufficiently strong meta-theory, which then means that there is a core that will always remain circular. And there's an interesting philosophical question of whether $PA$ is completely meaningful. Certainly we have empirically verified it for 'random' natural numbers up to way beyond $2^{1024}$ (such as in RSA), but the question is whether it 'holds' all the way for the real world, if not even our formal systems themselves become unjustifiable! – user21820 Mar 07 '16 at 06:14
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    @user40276: The icing on the cake is that Godel's incompleteness theorem tells us that no recursive extension of $PA$ is complete, but it crucially relies on addition and multiplication, which in particular means that no recursive extension $T$ of $PA$ can prove $Con(T)$, which on the surface of it states that $T$ is consistent. Ordinarily people simply treat this as inevitable, or argue that $Con(T)$ is actually meaningless, but either way $PA$ cannot understand its own consistency. However, https://en.wikipedia.org/wiki/Self-verifying_theories resembling $PA$ have been found! – user21820 Mar 07 '16 at 06:20

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First we have to choose a meta-system to work in, in which any formal system that we wish to study is just an object. Many meta-systems are possible, but they must all have something equivalent to string manipulation and induction. Traditionally people choose a meta-system that has the same proving strength as ZFC, simply because much of modern mathematics can be expressed in ZFC. However, ZFC is way stronger than needed for almost all results in logic. So some people choose a different meta-system such as some kind of type theory.

Next we define a formal system as a language over some alphabet together with a collection of rules describing which strings can be derived. Note that the alphabet, language, rules and strings are all objects in the meta-system. When we reason about a formal system we are not in the formal system, and can talk about what strings it can and cannot derive, whereas in the formal system all we can do is to follow the rules to produce more derived strings.

If you use a ZFC-equivalent meta-system, for example, you would say that a formal system consists of a set of strings that is generated by the rules applied to the empty set. (Some of the rules might say that you can always derive certain strings, which are called axioms.) This generation process can be defined using induction and then you need some further tool in the meta-system to collect all the strings that can be derived in $k$ steps over all natural numbers $k$. This would be trivial in any ZFC-equivalent meta-system because we can take $\bigcup_{k\in\mathbb{N}} S_k$ where $S_k$ is the set of strings that can be derived in exactly $k$ steps. But many other meta-systems can do the same too, so you are not at all restricted to using a set theory for your meta-system.

Note that very often you will need the law of excluded middle in the meta-system. For instance if you want to prove that for any first-order theory $T$ either $T + φ$ or $T + \neg φ$ is consistent, you will necessarily have to invoke excluded middle or equivalent at some point. This is true even if $T$ is intuitionistic! In general, most logicians would have no qualms saying that either a formal system can prove a certain formula or it cannot, which already requires the meta-system to have excluded middle. Also note that because of this, some constructions that you perform in the meta-system will not be computable.

But still in many cases, the theorems you prove in your meta-system actually have some concrete representation in the real world (or so it seems so far), as programs in some generic programming language. For example:

(For any recursive first-order theory)

  1. There is a program to check whether the input (string) is a well-formed formula.
  2. There is a program to check whether the first input is a valid proof of the second input.
  3. There is a program that outputs a proof of the input if the proof exists (if not it may not halt).

Even better:

There is a program that, given any proof checker program (for any arbitrary formal system $T$) and string $φ$, outputs a $\Sigma_1$-sentence over PA that is true (under the standard interpretation) if and only if $φ$ is provable in $T$.

The above is true despite $T$ being unknown to the programmer, not to say the program (which is only given the proof checker program)! Furthermore, for some formal systems $T$, this is the best possible, as there may not be a program that always outputs "yes" or "no" as the correct answer to whether $φ$ is provable in $T$. Of course, for any single $φ$ there is a program that outputs the correct answer even if we cannot figure out what that program is. But there is no single program that can get the right answer or any arbitrary $T$ (given as a proof checker program) and string $φ$!

user21820
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  • I felt like answering your question to address the underlying issues, so if you have further questions feel free to leave a comment. If you still want to see a relatively rigorous development of logic in set theory, you might try http://www.mcmp.philosophie.uni-muenchen.de/students/math/math_logic_munich.pdf. – user21820 Mar 07 '16 at 07:29