Abstract
Before we get into details regarding number structures, we will examine definability in cases that are easier to analyze. We define two important classes of structures: minimal, in Definition 9.1, and order-minimal, in Definition 9.4. The important concepts of type and symmetry were already introduced in Chap. 2; here we define them in general model-theoretic terms and use them to analyze the orderings of the sets of natural numbers, integers, and rationals.
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Notes
- 1.
Symmetries of structures are known in mathematics as automorphisms.
- 2.
A reminder: when we just say definable, we mean definable without parameters in the original language of the structure. If parameters are involved we say parametrically definable.
- 3.
In model theory, order-minimal structures are called o-minimal.
- 4.
This should not be confused with the natural notion of distance for rational numbers. It is definable in \(({\mathbb {Q}},+,<)\), but not in \(({\mathbb {Q}},<)\).
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Kossak, R. (2018). Minimal and Order-Minimal Structures. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_9
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DOI: https://doi.org/10.1007/978-3-319-97298-5_9
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