Abstract
Model theory is a main branch of applied mathematical logic. Here the techniques developed in mathematical logic are combined with construction methods of other areas (such as algebra and analysis) to their mutual benefit. The following demonstrations can provide only a first glimpse in this respect, a deeper understanding being gained, for instance, from [CK] or [Ho]. For further-ranging topics, such as saturated models, stability theory, and the model theory of languages other than first-order, we refer to the special literature, [Bue], [Mar], [Pz], [Rot], [Sa], [She].
The theorems of Lwenheim and Skolem were first formulated in the generality given in 5.1 by Tarski. These and the compactness theorem form the basis of model theory, a now wide-ranging discipline that arose around 1950. Key concepts of model theory are elementary equivalence and elementary extension. These not only are interesting in themselves but also have multiple applications to model constructions in set theory, nonstandard analysis, algebra, geometry and elsewhere.
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Notes
- 1.
All these conditions are also equivalent (they all hold) if the inconsistent theory is taken to be complete, which is not the case here, as we agreed upon in 3.3.
- 2.
This assumption is equivalent to the assertion that \(\{\gamma {\in }^{\langle X\rangle }\mid \mathcal{A}\vDash \gamma \}\) is complete; see the subsequent proof. For refinements of the theorem we refer to [HR].
- 3.
The “distance” d(a, b) between elements a, b of some SO-model is 0 for a = b, 1 + the number of elements between a and b if it is finite, and d(a, b) = ∞ otherwise.
- 4.
Moreover, the theory of all linear orders is decidable (Ehrenfeucht), and thus each of its finite extensions; but the proof is incomparably more difficult than for DO or SO.
- 5.
For uncountable \(\mathcal{A}\) we have \(\vert \mathcal{L}A\vert = \vert \mathcal{A}\vert \). In this case one proceeds with an ordinal enumeration of \(\mathcal{L}A\) rather than an ordinary one. But the proof is almost the same.
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Rautenberg, W. (2010). Elements of Model Theory. In: A Concise Introduction to Mathematical Logic. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1221-3_5
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