Abstract
This chapter serves as an interlude. Our goal in the following chapters is to show how tools of logic can used to uncover essential features of familiar number structures. We will be inspecting number structures with our logic glasses on, but we need to get used to wearing those glasses. In this chapter we will take a look at some easy to visualize structures—finite graphs—and we will examine them from the logical perspective. In other words, later, logic will help us to see structures; now, simple structures will help us to see logic. A concept of symmetry of a graph is introduced in Definition 2.1 followed by Theorem 2.1, which is crucial for many further developments. Both, the definition and the theorem, will be generalized later to arbitrary mathematical structures.
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Notes
- 1.
Notice the change of font. Throughout the book we will discuss various mathematical objects and for brevity we give them names, which are typically just single letters, sometimes using different fonts. There is nothing formal about those names, and choices of names are arbitrary.
- 2.
The Ehrenfeucht-Fraïssé theorem gives a criterion for determining the least quantifier complexity of first-order sentences that express differences between structures. See [47, Lemma 2.4.9].
- 3.
In each case, one of those symmetries is the identity transformation—nothing gets moved.
- 4.
It is now necessary to include \(\lnot (x_1=x_2)\) because we do not allow loops, so this is implied by \(E(x_1,x_2)\).
- 5.
Every graph has the trivial symmetry, i.e., the symmetry f such that \(f(v)=v\) for each vertex v.
References
Marker, D. (2002). Model theory: An introduction.Graduate Texts in Mathematics (Vol. 217). Springer-Verlag.
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Kossak, R. (2024). Logical Seeing. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-031-56215-0_2
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