Let Σ1 and Σ2 be sets of sentences such that nothing is a model of both Σ1 and Σ2. Show that the

step 1 Hello, so since nothing is a model of both Sigma 1 and Sigma 2, we have that Sigma 1, Union, Sig

Let Σ1 and Σ2 be sets of sentences such that nothing is a...

Let $\Sigma_1$ and $\Sigma_2$ be sets of sentences such that nothing is a model of both $\Sigma_1$ and $\Sigma_2$. Show that there is a sentence $\tau$ such that $\operatorname{Mod} \Sigma_1 \subseteq \operatorname{Mod} \tau$ and $\operatorname{Mod} \Sigma_2 \subseteq \operatorname{Mod} \neg \tau$. (This can be stated: Disjoint $\mathrm{EC}_{\Delta}$ classes can be separated by an EC class.) Suggestion: $\Sigma_1 \cup \Sigma_2$ is unsatisfiable; apply compactness.

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Key Concepts

Separation of Model Classes by a Sentence

Separating model classes by a sentence refers to the ability to find a single sentence that distinguishes between two classes of structures, assigning one class as models of the sentence and the other as models of its negation. This concept is important in model theory because it allows for clear semantic demarcation between theories or collections of models, enabling a deeper understanding of their structural and logical differences.

Unsatisfiability

Unsatisfiability is the property of a set of sentences wherein no model exists that can satisfy every sentence in the set simultaneously. When two sets of sentences are such that their union is unsatisfiable, it implies that they have no common model. This concept is crucial in arguments where the lack of a joint model is leveraged to establish separability between different classes of models.

Compactness Theorem

The Compactness Theorem is a fundamental result in first?order logic that asserts if every finite subset of a set of sentences is satisfiable, then the whole set is satisfiable. This theorem is often used to transfer properties from finite parts of a theory to the entire theory, making it an essential tool for proving the existence of models or sentences with desired properties.

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