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Russell's paradox and proper classes

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One of the most interesting paradoxes in set theory is known as Russell’s paradox. This paradox refers to the impossibility of speaking coherently of a set of all sets. But why can there not be a set of all sets? Because in conventional set theory, a set cannot be a member of itself. If we attempted to speak of a set of all sets R, this set would have to include itself among its members. But it cannot. Therefore, the supposed set of all sets is simply, as Steinhart points out, the set of all self-excluding sets.

Let R be the set of all sets. No set is a member of itself. Set x, which is a member of R, therefore, is not itself a member of x, since no sets are members of themselves. Since this criteria must apply to R itself, R cannot be included in itself as a member of the set of all sets. But in such a case, R as a meta-set would no longer be the set of all sets. Logically, this would mean that R would only be the set of all sets if it does not include itself, such that it is no longer the set of all sets. So in order for it to be the set of all sets, it must not be the set of all sets, which is of course logically absurd.

It seems intuitively unreasonable to deny that we can speak of some sort of group of all sets. Therefore, mathematicians have defined a specific group which would grant us such an inclusion. Rather than defining all groups as sets, some groups are defined as ‘classes.’ All collections as understood as forms of classes. Classes are either sets or proper classes. Proper Classes and sets are both subsumed under the broader concept of a ‘class.’ “Every set is a class, but not all classes are sets”(Steinhart, p. 63).

A set is a member of another class. A proper class is not. It stands on its own. X is a set iff there is some broader class to which it belongs. If there is no broader class to which this collection belongs, it is a proper class rather than a set. The class of all sets is represented V = {x|x is a set}. This solves Russell’s paradox because this collection of all sets is no longer itself a set, but is defined as a proper class.

Thus, the intuition that every property has an extension is preserved (Steinhart, p. 63). “The extension of a property is the collection of all and only those things that have the property. For instance, the extension of the property is-a-dog is the set {x|x is a dog}”(Steinhart, p. 62). Steinhart summarizes:

“…V is just the collection R from Russell’s Paradox. But now it isn’t paradoxical. Since V is a proper class, and not a set, V is not a member of V. After all, V only includes sets. Class theory thus avoids Russell’s paradox while preserving the intuition that every property has an extension. Class theory is more general than set theory. For every property, the extension of that property is a class.

A good way to understand the difference between sets and proper classes is to think of it in terms of the hierarchy of sets. A class is a set if there is some partial universe of V in which it appears…To say that a class appears in some partial universe is to say that there is some maximum complexity of its members. A class is a set if there is a cap on the complexity of its members.

A class is a proper class if there is no partial universe in which it appears”(Steinhart, p. 63)

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.