Proper class: Difference between revisions
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In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper class, and the [[Burali–Forti paradox]] shows that \(\mathrm{Ord}\). Also, all inner models, such as \(L\), are proper classes. |
In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper class, and the [[Burali–Forti paradox]] shows that \(\mathrm{Ord}\). Also, all inner models, such as \(L\), are proper classes. |
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Latest revision as of 16:51, 25 March 2024
In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper class, and the Burali–Forti paradox shows that \(\mathrm{Ord}\). Also, all inner models, such as \(L\), are proper classes.
Under the equivalence class definition of natural numbers, ordinals and cardinals, all numbers (other than 0) are proper classes, which is one of the downsides of this method.
ZFC is a strictly first-order theory, and thus a true treatment of proper classes is not possible within it. Instead, one only uses definable classes such as \(V\), \(\mathrm{Ord}\), and uses \(x \in X\) as a shorthand for \(\varphi(x)\), where \(\varphi\) is a first-order theory. Working within a true second-order treatment of proper, including non-definable, classes is a very powerful tool and can prove, for example, the existence of a proper class of worldly cardinals.[1]