Set theory as Ontological Foundation ...

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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