Russell’s Paradox

Singapore 3.13 AM 09.10.2015

Before I come to tell you about this incredibly famous paradox, I have to give you some sorts of basic definition about what is a paradox.

A paradox is “apparently unacceptable conclusion which is derived by an apparently acceptable reasoning from an apparently acceptable premises”

Ok let’s come back to famous Russell’s Paradox.

A set is any collection of things. A set’s members are the things in that set. As easy as it is.

We 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 be clearer, for example, we have

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 containg 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, that set of 4 sets is also a member of the set containing all the sets that have more than three members. In other word, that set of all of the set is the member of itself. Therefore, we call that type of set is a self-membered set.

So, 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 a elephant.


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

If S is a member of itself, it is imposible because set S is the set of all the non-self-membered, that means S, itself is a non-self-memberd set.

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

SO HOW????????

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