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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.
Barnes built finitism into the core of his approach to social theory and to the analysis of social order.
Of course, an analysis of the notion of finitism cannot be presented as a theorem.

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