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For an $L$-structure $\mathcal{A}$, the $L$-theory of $\mathcal{A}$ is the set of $L$-sentences: $$ \mathrm{Th}_L(\mathcal A) = \{\sigma : \mathcal A \models \sigma\} $$

Prove that $\mathrm{Th}_L(\mathcal A)$ is complete.

Whys is this true? Why can't there be an $L$-sentence such that it is neither true or false?

bof
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1 Answers1

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Assume there's an $L$-sentence $\sigma$ that's neither true nor false. Then we have that $\mathcal{A}\nvDash\sigma$ and $\mathcal{A}\nvDash\neg\sigma$. But this is a contradiction since $\mathcal{A}\vDash\sigma$ if and only if $\mathcal{A}\nvDash\neg\sigma$, and vice versa, by definition of what a structure is.

quanticbolt
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  • I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks! – bof Dec 08 '18 at 16:26
  • For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about). – quanticbolt Dec 08 '18 at 16:39
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    @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about). – Noah Schweber Dec 08 '18 at 16:40
  • @bof For any sentence $\varphi$ and any structure $A$ the definition of $A\models \lnot \varphi$ is that $A\not\models \varphi$. It immediately follows that either $A\models \varphi$ or $A\models \lnot \varphi$. – Alex Kruckman Dec 08 '18 at 18:34
  • thank you everyone – bof Dec 08 '18 at 19:27