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)
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)
This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in 2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Re-cursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERNA proves the main axioms of the Chuaqui and Sup-pes system.… (More)
We use model-theoretic methods described in  to obtain ordinal analyses of a number of theories of first-and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ 0 .
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)
Since 1985 the Education Program for Gifted Youth (EPGY) at Stanford University has been developing a series of stand-alone multi-media computer-based distance-learning mathematics and physics courses from the elementary school through the university level. Because these courses are used in situations where students do not have access to regular classroom… (More)
The field of computer science suffers from a lack of diversity. The Stanford Artificial Intelligence Laboratory's Outreach Summer (SAILORS), a two-week non-residential free summer program, recruits high school girls to computer science, specifically to Artificial Intelligence (AI). The program was organized by graduate student and professor volunteers. The… (More)