nonfinitist

Nonfinitist is a term used in the philosophy of mathematics and foundations of mathematics to describe advocates of non-finitist viewpoints. A nonfinitist rejects finitism, the view that mathematical objects must be finite in principle or that only finitary methods provide justification for mathematical truths. In contrast, nonfinitists accept the legitimacy of infinite objects, infinite processes, or completed infinite totalities as part of mathematical ontology or justification. They argue that certain mathematical theories—such as standard set theory with actual infinities, analysis including real numbers, and many proofs—cannot be adequately captured by finitary reasoning alone.

The term is used descriptively rather than as a single school, and many who accept actual infinities would be considered nonfinitists. It is distinct from intuitionism and constructivism, which also scrutinize the use of infinity but for different philosophical and methodological reasons. Nonfinitists may ground their view in platonic, formalist, or other philosophical commitments that permit or require infinite entities.

In practice, nonfinitist positions underlie most of contemporary mainstream mathematics, including Zermelo-Fraenkel set theory with the axiom of choice and classical analysis. They influence debates about the foundations of mathematics, the nature of mathematical proof, and the status of infinite objects within formal theories.

Related terms include finitism, actual infinity, potential infinity, and Cantor’s theory of infinity.