How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words-thousand, million, billion-to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and response time are expected near the boundaries between categories. In contrast to previous work (Landy, Silbert, & Goldin, 2013) that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers.

This article applies mathematical logic to obtain a rigorous foundation for previous inherently nonrigorous results and also extends those previous results. Roughly speaking, our main theorem states: any agent A that comprehends the correctness-related properties of software S also comprehends an intelligence-related limitation of S. The theorem treats the output of S, if any, as an attempt at solving a halting problem. Previous nonrigorous attempts to obtain similar theorems depend on infallibility assumptions on both the agent and the software. The hypothesis that intelligent agents and intelligent software must be infallible has been widely questioned. In addition, recent work by others has determined that well-known previous attempts use a fallacious form of reasoning; that is, the same form of reasoning can yield paradoxical results. Our main theorem avoids infallibility assumptions on both the agent and the software. In addition, our proof is rigorous, in the sense that in principle one can carry it out in Zermelo-Fraenkel set theory. The software correctness framework considered in the main theorem is that of Hoare logic.

Problem-solving software that is not-necessarily infallible is central to AI. Such software whose correctness and incorrectness properties are deducible by agents is an issue at the foundations of AI. The Comprehensibility Theorem, which appeared in a journal for specialists in formal mathematical logic, might provide a limitation concerning this issue and might be applicable to any agents, regardless of whether the agents are artificial or natural. The present article, aimed at researchers interested in the foundations of AI, addresses many questions related to that theorem, including differences between it and results of Gödel and Turing that have sometimes played key roles in Minds and Machines articles. This study also suggests that-if one is willing to assume a thesis due to Donald Knuth-the Comprehensibility Theorem is the first mathematical theorem implying the impossibility of any AI agent or natural agent-including a not-necessarily infallible human agent-satisfying a rigorous and deductive interpretation of the self-comprehensibility challenge. Some have pointed out the difficulty of self-comprehensibility, even according to presumably a less rigorous interpretation. This includes Socrates, who considered it to be among the most important of intellectual tasks. Selfcomprehensibility in some form might be essential for a kind of self-reflection useful for self-improvement that might enable some agents to increase their success. We use the methods of applied mathematics, rather than philosophy, although some topics considered could be of interest to philosophers.

The multiway rendezvous is a natural generalization of the rendezvous in which more than two processes may participate. The utility of the multiway rendezvous is illustrated by solutions to a variety of problems. To make their simplicity apparent, these solutions are written using a construct tailor-made to support the multiway rendezvous. The degree of support for multiway rendezvous applications by several well-known languages that support the two-way rendezvous is examined. Since such support for the multiway rendezvous is found to be inadequate, well-integrated extensions to these languages are considered that would help provide such support.

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