# Russell's Paradox

Can you solve this paradox? First thing first, what is a paradox? A paradox is "an apparently unacceptable conclusion which is derived...

Can you solve this paradox?

First thing first, what is a paradox? A paradox is "an apparently unacceptable conclusion which is derived by an apparently acceptable reasoning from an apparently acceptable premise"

We will begin with a very simple mathematical definition which is not unfamiliar if you complete your secondary education.

A set is any collection of things. Members of a set are the things that are in that set. As easy peasy as it seems. For example, we have a set of all the integer numbers "0,1,2,3,..,etc." The number "2" is considered a member of the set of all the integer numbers.

So far so good. We also have a notion that a set is a member of itself. For example, the set of all sets that have more than three members is a member of itself, since there are more than three sets that have more than three members.

To make you less confused, we have those following examples.

1. The set of 4 rulers is the member of the set containing all sets that have more than three members.

2. The set of 4 nations is the member of the set containing all sets that have more than three members.

3. The set of 4 students is the member of the set containing all sets that have more than three members.

4. The set of 4 computers is the member of the set containing all sets that have more than three members.

In total, we have 4 sets right? This set of 4 sets is also a member of the set containing all the sets that have more than three members. In other words, that set of all the sets that have more than three members is the member of itself. Therefore, we can call that type of set is

**self-membered**set.What about a non-self-membered set? The set of all elephants is not the member of itself since a set of all elephants is not an elephant. Another example is that a set of all the Vietnamese mothers is not the member of itself since a set of all the Vietnamese mothers is not a mother.

So here comes the paradox.

If we consider the set of all the non-self-membered set and we call this set is set S. Is S a member of itself?

If S is a member of itself. and we have set S is the set of all non-self-membered set. This leads to S is a non-self-membered set. This is contradictory because S is both a self-membered and a non-self-membered set.

If S is not a member of itself, therefore it is a member of the non-self-membered, but if S is a member of the non-self-membered, it has to be self-membered set because set S is the set of all non-self-membered. Again, set S is both self-membered and non-self-membered.

First question, do you understand the paradox?

Second question, can you solve it?

Second question, can you solve it?

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