Enderton 2.6.9

Say that a set $\displaystyle \Sigma$ of sentences has thefinite model propertyiff each member $\displaystyle \sigma$ of $\displaystyle \Sigma$, if it has any model at all, has a finite model. Assume that $\displaystyle \Sigma$ is a set of sentences in a finite language (i.e., a language with finitely many parameters) and that $\displaystyle \Sigma$ has finite model property. Give an effective procedure that, given any member $\displaystyle \sigma$ of $\displaystyle \Sigma$, will decide whether or not $\displaystyle \sigma$ has any models. Suggestion: Is the set of such sentences effectively enumerable? Is its complement effectively enumerable?

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(*) For a finite language, $\displaystyle \{\sigma | \sigma \text{ has a finite model}\}$ is effectively enumerable.

If $\displaystyle \sigma \in \Sigma$ has any models, then $\displaystyle \Sigma$ has a finite model. By (*), there is an effective procedure to produce the answer "Yes" if $\displaystyle \sigma$ has any models.

An effective procedure to produce the answer "No" if $\displaystyle \sigma$ does not have any models is as follows.

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Any help will be appreciated.