Sequences of cantor type and their expressibility

This paper deals with a criterion for sequences of Cantor type to have an expressible set with zero Lebesgue measure.

On expressible sets and p-adic numbers

Continuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

Some General Remarks on Provability and Truth

We have given three different incompleteness proofs of Peano Arithmetic— the first used Tarski’s truth-set, the second (Gödel’s original proof) was based on the assumption of ω-consistency, and the third (Rosser’s proof) was based on the assumption of simple consistency. The three proofs yield different generalizations—namely 1. Every axiomatizable subsystem of N is incomplete. 2. Every axiomatizable ω-consistent system in which all true Σ0-sentences are provable is incomplete. 3. Every axiomatizable simply consistent extension of (R) is incomplete. The first of the three proofs is by far the simplest and we are surprised that it has not appeared in more textbooks. Of course, it can be criticized on the grounds that it is not formalizable in arithmetic (since the truth set is not expressible in arithmetic), but this should be taken with some reservations in light of Askanas’ theorem, which we will discuss a bit later. It is not too surprising that Peano Arithmetic is incomplete because the scheme of mathematical induction does not really express the full force of mathematical induction. The true principle of mathematical induction is that for any set A of natural numbers, if A contains 0 and A is closed under the successor function (such a set A is sometimes called an inductive set), then A contains all natural numbers. Now, there are non-denumerably many sets of natural numbers but only denumerably many formulas in the language LA and, hence, there are only denumerably many expressible sets of LA- Therefore, the formal axiom scheme of induction for P.A. guarantees only that for every expressible set A, if A is inductive, then A contains all natural numbers. To express the principle of mathematical induction fully, we need second order arithmetic in which we take set and relational variables and quantify over sets and relations of natural numbers.

Laboratory Evaluation of a Unified Theory for Simultaneous Multiple Axis Artificial Arm Control

This paper reports on the application of a “postulate-based” control method for multi-axis artificial arm control. This method uses shoulder muscle EMG’s as control sites, but, unlike previous techniques, the theory is the first that can be rigorously defined in terms of musculoskeletal anatomy, EMG muscle-force relationships, EMG transmission characteristics, muscle recruitment, limb dynamics and normal motion constraints. The control theory results in a deterministic, mathematically expressible set of controller equations, which use the vector of natural limb torques estimated by shoulder EMG signals and a “constraint” for input. The output of the controller equations is a vector of prosthetic torques to be applied to the artificial limb. We report on the implementation of the theory up to the point of laboratory feasibility trials of actual simultaneous above-elbow amputee control of elbow flexion and humeral rotation. Implementation of the theory required: 1) deviation of the controller equations from Newton’s dynamic equations of motion into controller form in conformity with the postulate theory; 2) development of a methodology for estimating natural musculoskeletal torques from EMG signals; 3) hardware and software for experimental testing with actual closed loop amputee control of the prosthesis; and 4) a methodology for evaluating the performance of the prosthesis relative to both alternative prosthetic systems and the natural arm. These tasks were completed and simultaneous multiple-axis control of a prosthetic arm was accomplished by both amputee and nonamputee subjects. Key questions of control compatibility, naturalness, stability, and performance evaluation relative to other prostheses and the natural arm were addressed. Various problems are discussed with the conclusion that this method, despite some difficulties, holds great promise as a practical rehabilitation tool.