Wayne Aitken

Counterexamples to the Hasse Principle

Abstract This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexamples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate–Shafarevich groups.

Computer Implication and the Curry Paradox

There are theoretical limitations to what can be implemented by a computer program. In this paper we are concerned with a limitation on the strength of computer implemented deduction. We use a version of the Curry paradox to arrive at this limitation.

Abstraction in Algorithmic Logic

We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this situation leaves one no choice but to restrict the use of a certain class of implicational rules including modus ponens.

STABILITY AND PARADOX IN ALGORITHMIC LOGIC

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally.

A Note on the Physical Possibility of Transfinite Computation

In this note, we consider constraints on the physical possibility of transfinite Turing machines that arise from how one models the continuous structure of space and time in ones best physical theories. We conclude by suggesting a version of Churchs thesis appropriate as an upper bound for physical computation given how space and time are modeled on our current physical theories.

An explicit sign formula for the determinant of cohomology

The natural or “cohomological bases” construction of the determinant of cohomology functor has the problem that a certain diagram that we might want to commute only commutes up to sign, and rinding the value of this sign is a non-trivial problem. The more sophisticated determinant of cohomology functor of Knudson and Mumford has the property that the corresponding diagram commutes, not just up to sign. However, their construction leaves open the question of the sign behavior for the cohomological bases construction The cohomological bases construction has the advantage of being natural, concrete, and well-adapted to studying sections of determinant of cohomology sheaves that arise from choices of bases of relative cohomology sheaves (on Zariski open sets where the sheaves in question are free). Knudson and Mumford’s construction does not consider such sections, and, paradoxically, when one tries to define and study such sections with their constructions, one is again confronted with sign troubles. Being a...