Franklin's Notes

Realizing and omitting sets of formulas

If $\Sigma$ is a set of formulas in the free variables $x_1,\cdots,x_n$ over the language $\mathscr{L}$, and $\mathfrak{A}$ is a model of $\mathscr{L}$, then we say that $\mathfrak{A}$ realizes $\Sigma$ (or, $\Sigma$ is satisfiable in $\mathfrak{A}$) if some n-tuple of elements of $A$ satisfies $\Sigma$ (i.e. satisfies $\sigma$ for each $\sigma\in\Sigma$). On the other hand, we say that $\mathfrak{A}$ omits $\Sigma$ iff it does not realize $\Sigma$. Additionally, we say that $\Sigma$ is consistent iff it is satisfiable in some model.

For instance, the set of formulas $\Sigma(x)$ given by is omitted by the standard model of Peano Arithmetic but realized by the nonstandard models.



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