Let G := (V ,EG) be a 4-regular tree, i.e. an infinite connected undirected graph without cycles and such that each vertex has exactly 4 neighbors, the picture is a finite part of it. Recall that Th(G) denotes the theory of G, i.e. the set of all sentences that G satisfies. Using the Compactness theorem, prove that Th(G) has a disconnected model.

Let G := (V ,EG) be a 4-regular tree, i.e. an infinite connected undirected graph without cycles and such that each vertex has exactly 4 neighbors, the picture is a finite part of it. Recall that Th(G) denotes the theory of G, i.e. the set of all sentences that G satisfies. Using the Compactness theorem, prove that Th(G) has a disconnected model.

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Question

Let G := (V ,E^G) be a 4-regular tree, i.e. an infinite connected undirected graph without cycles and such that each vertex has exactly 4 neighbors, the picture is a finite part of it. Recall that Th(G) denotes the theory of G, i.e. the set of all sentences that G satisfies. Using the Compactness theorem, prove that Th(G) has a disconnected model.

Hint: Add two new constant symbols and sneakily make them farther and farther from each other.

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