Existance and example of extraordinary set and set of all such extraordinary sets in the context of Russels paradox

Question

I recently started studying elementary set theory on my own. I understood Russels paradox. By assuming set of all ordinary sets it leads to contradiction.
Now if i think about set of all extraordinary sets, then i got a doubt. Can you tell me an example of such "set of all extraordinary sets" and example for extraordinary set(set which contains itself).
I saw several posts in stackexchange about existence of extraordinary set(which say such set do not exist) and existence of set of all such extraordinary sets. But i did not understand. Pls explain in layman terms

If $X={ text{ any object x } :  text {x is not a Radio} }$
X is a set and so it is not a Radio and so X is inside X as an element. So can i say is this an extraordinary set?

Mauro ALLEGRANZA · Answer

It is more easy to reason in terms of "predicates", i.e. properties attributed to objects: humanity, color, etc.
Some (few) properties are predicable of itself: the property of being a predicate is predicable of itself. Most properties are not: humanity is not itself human. So, extraordinary predicates are predicates predicable of itself, while "common" predicates, that are not predicable of itself are ordinary ones. [Explanation due to B.Russell, 1903.]
In mathematics, sets are specific mathematical objects and they must not be considered as "sets" or "collections" of everyday life.
In particular, in mathematical theory of sets the principle that every predicate whatever specifies a set does not hold.
Due to "unnatural" nature of extraordinary predicates (see above) the mathematical theory of sets assumes that there are no extraordinary sets at all, i.e. all sets $A$ are "ordinary" ($A notin A$).
Having said that, regarding your example we have that $R = { x mid x text { is a radio } }$ is not a radio. Thus, $R∉R$ (ordinary).
But also $X = { x mid x text { is not a radio } }$ is not a radio; thus $X∈X$ (extraordinary).

Existance and example of extraordinary set and set of all such extraordinary sets in the context of Russels paradox

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