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Set theory as Ontological Foundation
Set theorists commonly take their subject to serve as a
foundation for the rest of mathematics, in the sense that other
abstract mathematical objects can be construed fundamentally
as sets, and in this way, they regard the set-theoretical universe
as the realm of all mathematics.
On this view, mathematical objects—such as functions, groups,
real numbers, ordinals—are sets, and being precise in
mathematics amounts to specifying an object in set theory.
A slightly weakened position remains compatible with
structuralism, where the set-theorist holds that sets provide
objects fulfilling the desired structural roles of mathematical
objects, which therefore can be viewed as sets.
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