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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?

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  • $\begingroup$ Set is a technical term with a technical definition which is hard to explain in layman's terms. It can be shown using the technical definition that no set can contain itself, that is, there are no extraordinary sets. So the set of all extraordinary sets is a bit like the set of all unicorns. $\endgroup$ – Gerry Myerson Jul 23 '20 at 13:27
  • $\begingroup$ @GerryMyerson sir pls have a look at example which i posted $\endgroup$ – Nascimento de Cos Jul 23 '20 at 13:34
  • $\begingroup$ If u can elaborate, it will help me in understanding things in a better manner. If possible, pls elaborate in your convinient time $\endgroup$ – Nascimento de Cos Jul 23 '20 at 13:35
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    $\begingroup$ 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.] $\endgroup$ – Mauro ALLEGRANZA Jul 23 '20 at 13:42
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    $\begingroup$ And yes : $R = \{ x \mid x \text { is a radio } \}$ is not a radio; thus, $R \notin R$ (ordinary), while $X = \{ x \mid x \text { is not a radio } \}$ is not a radio; thus $X \in X$ (extra). $\endgroup$ – Mauro ALLEGRANZA Jul 23 '20 at 13:50
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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).

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