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What is finitism?

What is finitism? Here are some definitions.

Noun
  1. (mathematics) An extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps.
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Examples
The make believe world of predicate logic was a mere conservative extension of finitism.
Of course, an analysis of the notion of finitism cannot be presented as a theorem.
Barnes built finitism into the core of his approach to social theory and to the analysis of social order.

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