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The principles of infinity arithmetic are explained. There is a need to bring J arithmetic and comparisons into line with these principles. Many of the results produced by J are mathematically anomalous and do not comply with the IEEE 754 Standard for floating-point arithmetic.Some proposals to generalise the concept of comparison tolerance are presented. There are also some tentative proposals regarding safety levels and precision.The paper is oriented towards J but many of the concepts are relevant to all dialects of APL.