Abstract
In this chapter we will see how one can learn something about a structure by using symmetries of its elementary extensions. We will examine the specific example of the ordering of the natural numbers, and we will prove that the structure \(({\mathbb {N}},<)\) is minimal. After so many pages, the reader will probably find it hard to believe that this example was my original motivation to write this book. Initially, it seemed that not much technical preparation was needed.
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Notes
- 1.
We are using the symbol + here. It is not in the language of \(({\mathbb {N}},<)\), but it is allowed as an abbreviation, since the expression φ(c + 1) can be written as ∀zS(c, z)⇒φ(z).
- 2.
This follows from a general fact that the number of pairs in an n-element set is \(\frac {n(n-1)}{2}\). For small values of n such as 6, one can verify it by listing all possible pairs.
- 3.
After British philosopher, mathematician, and economist Frank Plumpton Ramsey (1903–1930).
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Kossak, R. (2018). Elementary Extensions and Symmetries. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_12
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DOI: https://doi.org/10.1007/978-3-319-97298-5_12
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