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I'm trying to refresh my knowledge about mathematical logic and I'm still unsatisfied with my insight of Gödel's Completeness Theorem.

In my only "raid" into MathOverflow, I posed a similar question about Gödel's Completeness Theorem.

I received a good answer from Joel David Hamkins about this issue, but I was not sufficiently prepared to prosecute that discussion.

Gödel's proof is equivalent to Konig's Lemma [Gödel's paper says "it follows by familiar arguments" and a footnote by the editor comments : "Apparently by Konig's infinity lemma"].

R.Smullyan, into his First-Order Logic (1968), in his exposition of Compactness Theorem for propositional logic, refers to Konig's Lemma (pag.31); but he also points out (pag.34) "that although we used both analytic tableaux and Konig's lemma in our proof of the compactness theorem, neither is really essential". This is not clear to me ...

Question. Is it possible to "avoid" $\mathsf {ZF}$ and formalize Gödel's original proof only into f-o $\mathsf {PA}$ ?

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Mauro ALLEGRANZA
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  • Re q1, yes, but one has to be careful about how things are formulated. Perhaps it is better to work on $\mathsf{ACA}_0$, which is conservative over $\mathsf{PA}$ for arithmetic statements. This is discussed, for example, in Simpson's Subsystems of second order arithmetic. – Andrés E. Caicedo Jan 08 '14 at 20:21
  • The the bounty on the question actually makes me more hesitant to answer it. I think it might be better to wait longer before placing bounties. – Carl Mummert Jan 11 '14 at 12:00
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    I suspect that one reason you get few answers is that your questions are not phrased as just about the mathematics, they are thoroughly mixed with references to historical books and to older textbooks that people may not know. You also tend to ask several different questions at once. People prefer to be able to answer off the top of their head using their personal knowledge, rather than having to dig up old books. If you had just asked "Can Goedel's completeness theorem be formalized in Peano Arithmetic?" I suspect you would get an answer very quickly. – Carl Mummert Jan 11 '14 at 12:02
  • @Carl Mummert . I appreciate very much your suggestions, but I'm really interested into understanding "old books": may I specify in the header of the question that I'm asking for help understanding e.g."Smullyan's proof of ..." ? – Mauro ALLEGRANZA Jan 11 '14 at 15:58
  • @Mauro ALLEGRANZA Yes, you can certainly do that. But, in order to answer it, someone would have to be intimately familiar with the proof, or they would have to look up the book (if they even have it...) before they can answer. I think this may be a reason why you don't get too many answers to your questions. That is not to say that you can't ask them, but if you make them easier to answer by people who just know the mathematics, I think you will get more answers. – Carl Mummert Jan 11 '14 at 23:29
  • @Mauro ALLEGRANZA: I agree with Mummert. I cannot find the exact quotation you took from Kleene's book; perhaps I am reading a different edition of the book. Could you describe in modern terms the Theorem 36 that you see in Kleene's book, please? – Lawrence Wong Jan 12 '14 at 11:20
  • @Mauro ALLEGRANZA: I don't mean to disparage your historical questions; I am only speculating on why there are not many answers to them on this particular site. – Carl Mummert Jan 13 '14 at 14:01

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The completeness theorem is stated for arbitrary (even possibly uncountable) theories. If you restrict the theorem to only countable theories, then it is possible to prove the completeness theorem in weak subsystems of second-order arithmetic, such as $\mathsf{WKL}_0$, which has a consistency strength lower than PA.

The challenge with proving the completeness theorem in PA itself is stating the completeness theorem in PA itself. There is no good way to quantify over models in the language of first-order arithmetic, nor to quantify over arbitrary infinite theories (even countable ones).

I have no idea about Kleene's comment.

Carl Mummert
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  • Two more questions : (i) in saying that "the challenge with proving the completeness theorem in PA itself is stating the completeness theorem in PA itself", you are meaning that is not possible (or, up to now, no one found how) to do it ? (ii) about "weak subsystems of second-order arithmetic, such as $WKL0$", I think that I can find all the details in Simpson's book Subsystems of Second Order Arithmetic; is it too difficult ? (taking into account that - as you have seen - that I'm not mastering very well axiomatic set theory). – Mauro ALLEGRANZA Jan 13 '14 at 13:32
  • The problem is not that nobody knows how to state completeness in PA; it's that the language of PA is too impoverished to do it in any way that resembles what is usually called the completeness theorem ("a consistent first-order theory has a model"). On the other hand that can be stated in second-order arithmetic with only a mild restriction: any countable, consistent first-order theory has a countable model. As for Simpson's book, I am afraid it is quite advanced. Unfortunately there are no good undergraduate-level references on $\mathsf{WKL}_0$ that I know of. – Carl Mummert Jan 13 '14 at 14:01
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    A version of the Gödel Completeness Theorem holds for models of PA too. It is often called the Arithmetized Completeness Theorem, and goes back (I believe) to Hilbert and Bernays. As Mummert points out, it is not formulated in the language of PA. See, for example, Section 13.2 of Richard Kaye's Models of Peano Arithmetic, or Section I.4 of Hájek–Pudlák's Metamathematics of First-Order Arithmetic. – Lawrence Wong Jan 13 '14 at 17:06
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    @Lawrence -Kleene's reference (ISHI Press reprint) is on page 395 (Th.36) referred to as Hilbert-Bernays completeness theorem, 1939). I've found in Hájek–Pudlák's Metamathematics of F-O Arithmetic, Intro, pag.2 a reference to "Volume II of Hilbert-Bernays's monograph [Hilbert-Bernays 1939], containing a detailed exposition of arithmetization including the arithmetized completeness theorem." At pag.98 they say: "we prove a version of Godel's completeness theorem (called the Low arithmetized completeness theorem [Th.4.26, pag.104])." But I'm not able to reconcile the statements of the two. – Mauro ALLEGRANZA Jan 13 '14 at 20:55