Discovered in 1901 by Bertrand Russell, this paradox is an extremely intriguing contradiction which concerns the constraints of naïve set theory, and what we can define as a set. It can be represented informally as follows:
A set can be defined as a collection of distinct objects. Hence, there is nothing to say that a set which includes itself cannot exist. Thus, we can split all sets into two categories, normal and abnormal, where a normal set is one which does not include itself as an element, and an abnormal set is one which does.
Let C be the set of all normal sets. Suppose C is normal. Therefore, C must be included in the set of all normal sets, since it is normal itself. Hence C must be included within C. Therefore C must be abnormal.
Suppose C is abnormal. Therefore, since C is abnormal, it cannot be included in the set of all normal sets. Therefore, C cannot include itself. Therefore C is normal.
Therefore, if we assume C to be normal, it must be abnormal, and vice versa. So what is C? A set cannot be neither normal nor abnormal, since that means that the set neither contains itself, nor does not contain itself. In a similar fashion one can conclude that a set cannot be both normal and abnormal.
This paradox in fact lead to the change from naïve set theory, to the set theory we use today, known as ZermeloFraenkel set theory, or ZFC. In a nutshell, ZFC does not assume that for any given property, there exists a set of all sets which satisfies the property. And hence, we cannot assume such a set C exists.
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Avishek Dashttps://etonstem.com/author/avishekdas
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So does a set C exist, using the ZFC? Presumably if it does, the set would still follow Russell’s paradox?
ZFC does not assume that such a set C can exist for certain,or any set which contains all sets which meet a certain criteria for that matter. Instead, sets such as the set of all subsets of natural numbers, for example, do exist with ZFC.