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You can make calculus and Fourier transforms without infinite sets; all you need is limits.

• jlokier 5y ago
  Indeed you can. But you can also make calculus without limits, using explicit definitions of infinitesimals (called non-standard analysis or non-standard calculus), and in some ways this is simpler to use. You can approach it either way. As limits, or by extending the number system with particular consistent definitions of infinitesimals and infinities. In practice using infinitesimals is often what people actually do when using calculus anyway, even when mentioning the limit in a sloppy hand-wave in the background, so it's good that non-standard analysis shows that the simpler method people actually use is usually logically coherent and gives the same answers as limits.