# The Definition of Consistency and Compactness in FOL

First order logic: "consistency," "compactness"?

Consistency: A set $\Sigma\subseteq\text{WFF}$ is consistent iff there is no $\varphi\in\text{WFF}$ such that $\Sigma\vdash\varphi$ and $\Sigma\vdash\lnot\varphi$.

Compactness: A set $\Sigma$ is consistent iff every finite $\Sigma_0\subseteq\Sigma$ is consistent.

I have read through these definitions, however I still feel like I do not understand them. Can someone respond and break them down for me in simpler terms? I feel more confident in my abilities if I understand them to the fullest and their application.

Thank you

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What definitions? They are missing. – Guy Coder Dec 10 '12 at 1:31
I don't think that a general explanation of consistency and compactness is the sort of question that the FAQ for the site asks for. In particular, it is hard to see that there is a correct answer to the question, such as it is. The FAQ says, " If you can imagine an entire book that answers your question, you’re asking too much." For that reason, I voted to close. However, the question could be improved by asking about the specific thing that you would like to know about. – Carl Mummert Dec 10 '12 at 1:33
could you explain them further then the definitions? – Masterminder Dec 10 '12 at 1:35

The definition of a theory being consistent is that it can not prove both $\varphi$ and $\neg\varphi$. However an alternative way of think about a theory $T$ in the language $\mathcal{L}$ being consistent is that $T$ can not prove all $\mathcal{L}$ sentences. In this way an inconsistent theory is not very intersesting since $T$ can prove every sentence. Moreover, a theory being consistent also means that it has a model, i.e. a $\mathcal{L}$-structure $\mathcal{M}$ such that $\mathcal{M} \models \varphi$ for all $\varphi \in T$.

The compactness theorem of logic states that a theory $T$ is consistent if and only if every finite $\Lambda \subset T$ is consistent. Model theoretically, $T$ has a model if and only if every finite set of sentences of $T$ has a model. In practice, it's fairly easy for certain theories to construct models for any finite collection of sentences. Then using the compactness theorem, you can conclude the whole (possible infinite theory $T$) has a model (i.e. is consistent) without explicitly producing a model for $T$.

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