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Could someone please explain to me Gödel's Completeness Theorem, with a simple example?

Edit: I realize I might be asking for a lot, instead in the edited question: what are: Completeness, soundness and provability? I went through the wikipedia page on it, but its a little dense for me right now.Also, are there any assumptions associated with the completeness theorem?

novice
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  • You need to understand the following terms: (1) sentence; (2) theory; (3) proof; (4) structure; (5) the satisfaction relation. If you do that, then the completeness theorem is easy to explain. $T$ is a theory, then $T$ proves $\varphi$ if and only if every structure which satisfies $T$ also satisfies $\varphi$. (Of course you may need to know and understand auxiliary terms, e.g. sets, functions, etc.) – Asaf Karagila Jan 02 '15 at 11:51
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    Completeness theorems make a parallel between provability and truth. You might want to consider tackling a simpler completeness theorem, other than Gödel's Completess Theorem. For instance the completeness theorem for propositional calculus. – Git Gud Jan 02 '15 at 11:53

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Completness theorem states that:

If $\tau$ is a first-order-sentence such that $\tau$ is valid (true under Any intrpretation), then $\tau$ is provable from the axiomatic frame of the first order logic.

To understant this, It's helpful to remember that while studying logic, we make a distinction between the syntatic and the semantic perspectives. we deal with the synax and semantics seperately and then tie them together by soundess and completness theorems.

In the syntax, you have some set of formulas called "logical axioms" which is a fixed set. and a set of sentences called "non-logical axioms" which varies according to the topic you have in mind and some "rules of inference " which tells you which sentences you can deduce from which.

You have to know that, non-logical axioms varies as mentioned, for example, if you discuss group theory then you will pick some axioms diffrent from those of topological spaces. So, non-logical axioms determine the theory you wish to work on. logical axioms represtent the axioms that are Always valid i.e. no matter which subject you work on, they are always satisfied, logical axioms could be something like $(a=b) \& (b=c) \rightarrow a=c$ " transitivity of equality relation"

For rules of inferences, if you have the rule called modes ponens in your system, it tells you that you can deduce the sentence $\alpha$ from the sentences $\beta\rightarrow\alpha , \beta$.

Then you can construct deductions, those are lists of sentences. every sentence is either a logical axiom or non-logical axiom or a sentence that is deduced from some earlier sentences by means of a rule of inference. A sentence $\sigma$ is provable if there is a deduction whose last sentence is $\sigma$.

For the semantic part, this is how you interpret the symbols, for example, you can say that your symbols represnt natural numbers and that the symbol # represents addition of numbers and so on. Some sentences will be true undersome intrepretations and false under some others. Some will be true under ALL of the intrpretations "those are called, valid sentences" and some will be false under any interpretation.

Now, completness theorem says that, If you are given a sentence which is valid i.e. true under any interpretation, then you will find a deduction which ends up with the that sentence. What does that mean is the you will find a proof for every valid sentence.

FNH
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    @novice You're asking for too much. Not because it's a lot of work to get into all that, but because such an undertaking wouldn't be a good fit for MSE's Q&A. I think you'd do best in grabbing a book.. – Git Gud Jan 02 '15 at 12:06
  • @novice, An interpretation is giving "meaning" to symbols. Remember, syntax means only "manipulation" of symbols without any regard to the meaning of the symbols, so for example , $(x<y)$ is a forumla. When you interpret it, that is to give meaning to $x,y$ and to the relation $<$. So, you could interpret $x$ as $5$ , $y$ as $6$ and $<$ as "less than". But you could interpret $x$ as a person whose name is "Alfred" ,$y$ as "Adams" and "$<$" as "is a father of". So $x<y$ means, Alfred is a father of Adams. this is an interpretation, we gave a meaning to symbols. To be continued .. – FNH Jan 02 '15 at 15:17
  • So, any given meaning to the symbols counts as an interpretation, no matter how odd this meaning is, as long as it's not self-contradictiry. for example, for any variable,say x, you can only gives it a unique interpretation "meaning". You can't say that "$x$" refers to both , $5$ and $6$. or "adams" and "john". To sum up, to interpret is to assign meaning to the symbols. – FNH Jan 02 '15 at 15:19
  • @novice, I'm not a professional on the subject, I'm no more than an undergraduate student. You can look at this site "http://www.logicmatters.net/tyl/ " where you can find many many recommendations and reviews on books to study mathematical logic from. But, I myself have studied it from "A friendly introduction to mathematical logic, Leary" which deserves its name. But, it's out-of-print nowdays, If you like it, you can look for an old version or look on it in a library. – FNH Jan 02 '15 at 15:44