Abstract
The compactness theorem, Theorem 11.2, is one of the most frequently used basic tools of model theory. It implies that for every structure with an infinite domain there is another structure that is very similar but not isomorphic to the given one. We will see a toy example that shows how such structure could be used to study number-theoretic problems. A more advanced application is given in Appendix A.5.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For a sentence φ to logically follow from a set of sentences, means that there is a formal proof of φ in which sentences form the set are used as premises. Formal rules of proof have to be specified, and this can be done in several ways.
- 2.
In technical terms, the theory of a structure is axiomatizable if it has either finite, or a recursive (computable) set of axioms.
- 3.
Gödel proved the compactness theorem for countable languages. The theorem was extended to uncountable languages by Anatoly Maltsev in 1936.
- 4.
It is not the only formalism that is used in practice. A powerful alternative is category theory that is preferred in certain areas of algebra and geometry.
- 5.
For a very readable and comprehensive account see Simon Singh’s book [31].
- 6.
The parameters 0, 1, and 2 can be eliminated from these formal statements, since they are definable in each structure.
- 7.
Each structure is considered an extension of itself and it is an elementary extension. A proper extension is an extension that adds new elements to the domain of the extended structure.
- 8.
Larger in the sense of containment of sets, not their cardinalities.
- 9.
Continuum is the cardinal number of the set of real numbers \({\mathbb {R}}\).
- 10.
According to Wikipedia, the current largest twin prime pair known is 2996863034895 ⋅ 21290000 ± 1, with 388,342 decimal digits. It was discovered in September 2016.
- 11.
Some small alterations are needed to make TPC comply with the first-order formalism. To express TPC as a first-order sentence in the language of +  and ⋅, one has to replace <  by its definition in \(({\mathbb {N}},+,\cdot )\), the expression P(y + 2) can be written as ∀z[(z = y + 2)⇒P(z)], and the reference to 2 can be eliminated with the help of its definition in \(({\mathbb {N}},+,\cdot )\).
References
Hilbert, D. (1926). Uber das Unendliche. Mathematische Annalen, 95, 161–190.
Singh, S. (1997). Fermat’s last theorem. London: Fourth Estate.
Wittgenstein, L. (1976). Wittgenstein’s lectures on the foundations of mathematics, Cambridge, 1939 (Edited by Cora Diamond). Chicago: The University of Chicago Press.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG part of Springer Nature
About this chapter
Cite this chapter
Kossak, R. (2018). Where Do Structures Come From?. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-97298-5_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-97297-8
Online ISBN: 978-3-319-97298-5
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)