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Undecidability in Number Theory

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Model Theory in Algebra, Analysis and Arithmetic

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2111))

Abstract

In these lecture notes we give sketches of classical undecidability results in number theory, like Gödel’s first Incompleteness Theorem (that the first order theory of the integers in the language of rings is undecidable), Julia Robinson’s extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers, and Matiyasevich’s negative solution of Hilbert’s 10th problem, i.e., the undecidability of the existential first-order theory of the integers. As Hilbert’s 10th problem is still open for the rationals (i.e., the question whether the existential theory of the field of rational numbers is decidable) we also present a sketch of the fact that there is a universal definition of the ring of integers inside the field of rationals. In terms of complexity this is the simplest definition known so far. If one had an existential definition instead then Hilbert’s 10th problem over the rationals would reduce to that over the integers (and hence be, as expected, unsolvable), but, modulo a widely believed in conjecture in number theory, we also indicate why there should be no such existential definition. We conclude with a list of nice open questions in the area.

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Notes

  1. 1.

    Even though Tarski may be better known for his decidability results than his undecidability results, one should point out that his ‘Undefinability Theorem’ (1936) that arithmetical truth cannot be defined in arithmetic is very much in the spirit of Gödel’s Incompleteness Theorems (in fact, it was discovered independently by Gödel while proving these). Tarski also proved undecidability of various other first-order theories, like, e.g., abstract projective geometry. This may put our very rough initial picture of the five counterpointing pairs into a more accurate historical perspective.

  2. 2.

    The use of the terms ‘local’ and ‘global’ which one more typically encounters in analysis or in algebraic geometry hints at a deep analogy between number theory and algebraic geometry, more specifically between number fields (i.e. finite extensions of \(\mathbb{Q}\), the global fields of characteristic 0) and algebraic function fields in one variable over finite fields (i.e., finite extensions of the field \(\mathbb{F}_{p}(t)\) of rational functions over \(\mathbb{F}_{p}\), the global fields of positive characteristic). It is one of the big open problems in model theory whether or not the positive characteristic analogue of \(\mathbb{Q}_{p}\), i.e., the local field \(\mathbb{F}_{p}((t))\) of (formal) Laurent series over \(\mathbb{F}_{p}\), is decidable.

  3. 3.

    Sometimes the term p-adically closed refers, more generally, to fields elementarily equivalent to finite extensions of \(\mathbb{Q}_{p}\)—cf. [59].

  4. 4.

    That we write the first name only for the lady is neither gallantry nor sexism: the reason is that there are other Robinsons in the same area: Abraham Robinson, one of the founders of model theory, and the logician and number theorist Raphael Robinson, also Julia’s husband.

  5. 5.

    c K denotes the class number of K, that is, the size of the ideal class group of K. It measures how far \(\mathcal{O}_{K}\) is from being a PID: c K  = 1 iff \(\mathcal{O}_{K}\) is a PID, so c K  = 2 is ‘the next best’. It is not known whether there are infinitely many number fields with c K  = 1.

  6. 6.

    The set E(K) of K-rational points of E is a finitely generated abelian group isomorphic to the direct product of its torsion subgroup E tor (K) and a free abelian group of rank ‘rk \((E(K))\)’.

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  1. Mathematical Institute, Radcliff Observatory Quarter, Oxford, OX2 6GG, UK

    Jochen Koenigsmann

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Koenigsmann, J. (2014). Undecidability in Number Theory. In: Model Theory in Algebra, Analysis and Arithmetic. Lecture Notes in Mathematics(), vol 2111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54936-6_5

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