Abstract
In the previous chapter we saw examples of mathematical structures that are simple enough to allow a complete analysis of the parametrically definable subsets of their domains. Those structures are of some interest, but the real objects of study in mathematics are richer structures such as \(({\mathbb {Q}},+,\cdot )\) or \(({\mathbb {R}},+,\cdot )\). To talk about them we first need to take a closer look into their definable sets. Definable sets in each structure form a geometry in which the operations on sets are unions, intersections, complements, Cartesian products, and projections from higher to lower dimensions. We will see how those operations correspond in a natural way to Boolean connectives and quantifiers, and how the name “geometry” is justified when it is applied to sets definable in the field of real numbers. The last two sections are devoted to a discussion of the negative solution to Hilbert’s 10th problem.
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.
After George Boole (1815–1864).
- 2.
Recall that a unary relation is a subset of the domain of a structure.
- 3.
Notice that another special case ax 2 + by 2 = 0 the defined set is just {(0, 0)}. One point can be considered as a circle that has radius 0.
- 4.
The distance between two numbers a and b in \({\mathbb {R}}\) is defined to be \(\sqrt {(a-b)^2}\), which can also be defined more naturally as |a − b|.
- 5.
The other one was David Hilbert.
- 6.
If X is a metric, or in general a topological space, then the set of all Borel sets of X is the smallest set that contains all open subsets of X and is closed under complements, and countable unions and intersections i.e. if A is Borel, then so is X ∖ A, and if A 1, A 2… is a sequence of Borel sets, then the union and the intersection of all sets in that sequence are also Borel.
- 7.
A square is a number of the form n 2, and a cube is a number of the form n 3.
- 8.
It was known well before Wiles’ proof that the equation x 3 + y 3 = z 3 has no positive integer solutions. This was proved by another great mathematician Leonard Euler (1707–1783). Many other cases of Fermat’s Last Theorem had been proved before Wiles announced his result. The smallest exponent n for which the theorem had not been verified before, is 4,000,037. This last fact is not very well-known. I am grateful to my colleague Cormac O’Sullivan for digging it up.
- 9.
After the third century Greek mathematician Diophantus of Alexandria.
- 10.
For a very touching personal account of the history of the MRDP theorem see [29]. Technical details are included.
- 11.
In practice it may not be as simple as it appears, and in fact quite often it is not. The polynomial q(z 1, z 2, z 3, z 4, z 5, z 6) may be of high degree, and calculations needed to evaluate it may be tedious, or simply too hard, even for a fast computer.
References
Reid, C. (1996). Julia, a life in mathematics (With Contributions by Lisl Gaal, Martin Davis and Yuri Matijasevich). Washington, DC: Mathematical Association of America.
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). Geometry of Definable Sets. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-97298-5_10
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)