Finitism concerns the philosophy of mathematics that rejects the existence of uncountably infinite sets. Some finitists also reject countably infinite sets.

A rough taxonomy of finitists is as follows.

  • classical finitists, who reject only uncountably infinite sets.

  • strict finitists, who reject all infinite sets, including $\mathbb{N}$.

  • ultrafinitists, who reject all infinite sets, and additionally finite structures which are for any conceivable practical purpose.

Many finitists also identify as finitists in practice, meaning that although they acknowledge the existence of various infinite sets, they do not use them when doing mathematics.

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