Abstract
In this chapter we shall introduce various methods for the construction of models of an axiom system Σ of sentences in a formal language L. In Chapter 1 we have already encountered a method, in the form of the so-called term-models, for obtaining at least one model of Σ. Having presented that “absolute” construction, we shall now present a series of “relative” constructions. These relative methods allow us to start with one or more given models of Σ, and to produce a new model. The methods considered here do not (as often occurs in mathematics) depend on the particular axiom system Σ (e.g. the direct product of groups is again a group, while the analogue of this for fields does not hold); rather, our methods will work in every case. This will be guaranteed by the fact that an L-structure \(\mathfrak{A}'\) obtained by these methods from an L-structure \(\mathfrak{A}\) is elementarily equivalent to \(\mathfrak{A}\); i.e.
Therefore, if \(\mathfrak{A}\) is a model of Σ, then so is \(\mathfrak{A}'\), independent of which axiom system Σ⊆Sent(L) we are working with.
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© 2011 Springer-Verlag London Limited
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Prestel, A., Delzell, C.N. (2011). Model Constructions. In: Mathematical Logic and Model Theory. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2176-3_3
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DOI: https://doi.org/10.1007/978-1-4471-2176-3_3
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2175-6
Online ISBN: 978-1-4471-2176-3
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