Alternative foundational systems such as von Neumann–Bernays–Gödel (NBG) set theory formally accommodate proper classes, including the class of all cardinals, as legitimate mathematical objects with determinate extensions.

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Class(as used in set theory): A mathematical term for a collection or grouping of objects that share something in common.
Determinate extensions(as used in logic and mathematics): Having clear, definite boundaries that specify exactly which objects belong to the collection and which don't.
Foundational systems(as used in mathematics and logic): Basic mathematical frameworks that establish the rules and building blocks for all other math to rest on—think of them as the ground floor that everything else is built on top of.
Legitimate mathematical objects(as used in mathematics and philosophy of mathematics): Things that are officially recognized and accepted as real entities within a mathematical system that can be studied and reasoned about.
Proper classes(as used in set theory): In set theory, collections of things that are too large or too weird to be treated as ordinary 'sets' (think of them as super-big collections).
Set theory(as used in mathematics): A branch of mathematics that studies collections of objects (called 'sets') and the rules for how they relate to each other.
von Neumann–Bernays–Gödel (NBG) set theory(as used in mathematics): A formal mathematical system created by three mathematicians (John von Neumann, Paul Bernays, and Kurt Gödel) that describes how collections of objects can be organized and related to each other.

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