*1.2. Axiomatic Model Theory*

First-order model theory is also the most important example of the explicit and concrete approach to model theory. The axiomatic approach contrasts this as the concepts and defining properties are axiomatised rather than considered concretely. As with all other axiomatic approaches in mathematics, this achieves proper abstraction, relativisation, conceptual clarity, and structurally clean causality. In a sophisticated mathematical area such as model theory, these features are crucial. The very origins of the axiomatic approach

**Citation:** Diaconescu, R. The Axiomatic Approach to Non-Classical Model Theory. *Mathematics* **2022**, *10*, 3428. https://doi.org/10.3390/ math10193428

Academic Editor: Francesco Calimeri

Received: 6 August 2022 Accepted: 16 September 2022 Published: 21 September 2022

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to model theory may be traced back, although not in an explicit form, in Lindström's "external" characterisation of first-order logic [8]. Several explicit axiomatic developments followed, such as Barwise's *abstract model theory* [9,10] or the *categorical model theory* of the Budapest school [11–15], etc. In spite of their success in developing interesting results, all those approaches lacked full axiomatisability, as they would usually treat axiomatically some parts of the logical systems while considering concretely other parts. Consequently, they were not able to achieve the true power of the full axiomatic approach.
