The modal logic of arithmetic potentialism and the universal algorithm

I investigate the modal commitments of various conceptions of the philosophy of arithmetic potentialism. Specifically, I consider the natural potentialist systems arising from the models of arithmetic under their natural extension concepts, such as end-extensions, arbitrary extensions, conservative extensions and more. In these potentialist systems, I show, the propositional modal assertions that are valid with respect to all arithmetic assertions with parameters are exactly the assertions of S4. With respect to sentences, however, the validities of a model lie between S4 and S5, and these bounds are sharp in that there are models realizing both endpoints. For a model of arithmetic to validate S5 is precisely to fulfill the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal $\Sigma_1$ theory. The main S4 analysis makes fundamental use of the universal algorithm, of which this article provides a simplified, self-contained account. The paper concludes with a discussion of how the philosophical differences of several fundamentally different potentialist attitudes---linear inevitability, convergent potentialism and radical branching possibility---are expressed by their corresponding potentialist modal validities.Comment: 38 pages. Inquiries and commentary can be made at http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm. Version v3 has further minor revisions, including additional reference

On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a wide class of interesting extensions, including cases in which the full ETNC in not known. Similarly, we construct annihilators of higher dimensional algebraic K-groups of the ring of integers in L.Comment: 33 pages, error in section 3.4 corrected. To appear in Transactions of the AM

Many projectively unique polytopes

We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its f-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively unique polytopes, among them projectively unique polytopes inscribed to the sphere. The proofs rely on a novel construction technique for polytopes based on solving Cauchy problems for discrete conjugate nets in S^d, a new Alexandrov--van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence's extension technique for point configurations.Comment: 44 pages, 18 figures; to appear in Invent. mat

The growth of the rank of Abelian varieties upon extensions

Number theory, Algebra and Geometr