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The relative strengths of first-order theories axiomatized by transfinite induction, for ordinals less-than ~0, and formulas restricted in quantifier complexity, is determined. This is done, in part, by describing the provably recursive functions of such theories. Upper bounds for the provably recursive functions are obtained using model-theoretic(More)
Science serves its readers as a forum for the presentation and discussion of important issues related to the advance-The de ment of science, including the presentation of minority or ly condu conflicting points of view, rather than by publishing only .i. material on which a consensus has been reached. Accord-OlflOfl ingly, all articles published in(More)
Science serves its readers as a forum for the pi-esentation and discussion of important issues related to the advancement of science, including the presentation of minority or conflicting points of view, rather than by publishing only material on which a consensus has been reached. Accordingly , all articles published in Science-including editorials, news(More)
The EPGY Theorem-Proving Environment is designed to help students write ordinary mathematical proofs. The system, used in a selection of computer-based proof-intensive mathematics courses, allows students to easily input mathematical expressions, apply proof strategies , verify logical inference, and apply mathematical rules. Each course has its own(More)
We describe a model-theoretic approach to ordinal analysis via the finite com-binatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with com-binatorial properties that, in nonstandard instances, give rise to models of the theory being(More)
Simulation-based training is becoming an accepted tool for educating physicians before direct patient care. As ultrasound-guided regional anesthesia (UGRA) becomes a popular method for performing regional blocks, there is a need for learning the technical skills associated with the technique. Although simulator models do exist for learning UGRA, they either(More)