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Joel David Hamkins
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Consider the integers with their successor relation: $$\langle \mathbb{Z},\varepsilon\rangle,$$ where $n\mathrel{\varepsilon} (n+1)$ and these are the only instances of the relation. In this model, every set has exactly one element, its predecessor.

Since this structure has nontrivial automorphisms by translation, which move every point, it has no definable elements. It consequently satisfies vacuously all the definable axioms you mention. It also satisfies extensionality and foundation. This structure has no transitive sets and therefore no objects that would be finite von Neumann ordinals; so every object contains them all vacuously as elements, and thus it also satisfies this version of the infinity axiom.

So this theory does not interpret ZFC and the consistency strength is very weak.

Joel David Hamkins
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