The real number system can be enlarged to include infinitely large and infinitesimally small numbers. These notions were used by the founders of the Calculus, but were banished for two hundred years until it was discovered in 1960 how to make proper sense of them, using methods originating from mathematical logic. This half-paper will develop the conceptual foundations of nonstandard analysis, and discuss some of its unique principles of reasoning and applications to analysis, topology, measure theory etc.
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