Are Paradoxes really necessary?

What is a PARADOX?
A paradox is a self-contradictory statement, which can only be true if it is false, and vice versa. It is a claim that two apparently contradictory statements are true.

Paradoxes can be used anywhere. They are used in literature, philosophy, Science and paradoxes exist even in Mathematics. What is the need of using a paradox? If the statements are self-contradictory, why must we even spend so much time on learning them. Paradoxes, though are self-contradictory, they help us reveal the flaws either in the way it is stated or in the logics used to come to a conclusion based on those statements.

You have learnt about statements in mathematics or elsewhere. A statement is a kind of expression of something that is either TRUE or FALSE and cannot be both or neither. Consider this statement here,

p: This statement p is false.

Here, the statement says "statement p is false". So if the statement p is false then the implication made by it will be false and hence the statement cannot be false. So it must be true. But, if the statement is true, then what it says must be true which again brings us back to beginning and implies that "statement p is false". A bit confusing isn't it?? Well, don't panic, you are not alone. If you're seeing this for the first time it is normal to get confused. Another set of false statements can be as follows,

The statement below is true.

The statement above is false.

Here again, if the first statement is true then what it says must also be true and it says that the second statement is true. But, the latter statement states that the former statement is false and hence what the former states must also be false, which brings us back to the beginning loop of our deduction. No matter whether you consider the first statement to be true or the second statement, you will end up concluding nothing. 

But why do we require such paradoxes in Mathematics? 

It is because, they point out the flaws in definitions which intuitively seem to be true, and have caused axioms of mathematics and logic to be re-examined. One such famous, yet fundamental, is the Russell's Paradox which is also known as Russell's Antinomy, discovered by Bertrand Russell in 1901. Russell's Paradox  questions whether "set of all sets that contain themselves" would contain itself.

Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's Paradox.

There are many more paradoxes, some of which related to mathematics and many others which do not. Few that requires logic to understand and many others that do not need any special knowledge. This post today is just to ignite a spark of interest in the fresh minds to know more about Mathematics.

Comment your opinions and suggestions in the comment box if you need more clarification or whether I must simplify the complexity of language used in this post. I suggest you to discuss with your teachers and fellow classmates about these and learn many more paradoxes that exist, out there in the vast world of logic, assumptions, and whatsoever.

Following are few links to few other paradoxes you might want to take a look at:
1. The barber paradox
2. The paradox of Theseus' Ship
3. Zeno's paradox
4. The paradox of heap
5. The grandfather paradox