High Quality Content by WIKIPEDIA articles! In the philosophy of mathematics, ultrafinitism, or ultraintuitionism, is a form of finitism. Like other strict finitists, ultrafinitists deny the existence of the infinite set N of natural numbers, on the grounds that it can never be completed. In addition, ultrafinitists are concerned with our own physical restrictions in constructing (even finite) mathematical objects. Ultrafinitism is a form of constructivism, but even constructivists generally view the philosophy as unworkably extreme. The logical foundation of ultrafinitism is unclear; in his comprehensive survey Constructivism in Mathematics (1988), the constructive logician A. S. Troelstra dismissed it by saying "no satisfactory development exists at present." This was not so much a philosophical objection as it was an admission that, in a rigorous work of mathematical logic, there was simply nothing precise enough to include.

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