November 12, 2016 Saturday

Bedtime Story 


Troublesome Paradoxes in Mathematics

        

Most ordinary people and semi educated like me believe that there is nothing more consistent, self-contained that stands tall on a firm logical foundation than mathematics.

But mathematicians like David Hilbert knew better.

In the early 1900s, certain paradoxes kept popping up which went against the idea that the foundations of mathematics could be consistently stated within mathematics itself.

You make wonder what kind of paradox are we talking about?

Paradox is a statement that stems from apparently true premises that either contradicts itself or ends up in a conclusion that is logically unacceptable.

One must not mix up paradox with contradiction.

A contradiction in the mathematical formal system is a very useful tool to prove absurdity of a theorem or an assumption.

For example, if solution to any algebraic equation ends up proving that 10 – 9 = 2, then it serves a valid reason or even a proof of inconsistency of that equation.  

That calls for rejection and invalidity of the equation.

So a contradiction is a useful tool in mathematics to prove or disprove a theorem or assertion.

Paradox on the other hand upsets the applecart because it reveals an inherent inconsistency to which there is no way around.

One of the most famous paradox was discovered by both Ernst Zermelo and a year later by Bertrand Russell.   

They both discovered it while working on the naïve set theory and continuum hypothesis that was introduced and advanced by the great mathematician Georg Cantor in 1878.

Cantor’s continuum hypothesis is a landmark feat in the annals of mathematics that in order to be deal with would need a few bedtime stories.

Cantor’s continuum hypothesis and naïve set theory were informal in the sense that they used natural language to describe sets or any operations that was done on the sets.

Both Ernst Zermelo (in 1900) and Bertrand Russell (in 1901) when they tried to formalize the set theory, ran into a paradox.

The paradox is this.

A definable collection is a set.

Let Z be the set of all sets that are not members of themselves.

Then if Z is not a member of itself, then by definition it contains itself.

Yet, if it contains itself, it defies its own definition of sets that are not members of themselves.

In short:

Let Z = {x such that x is not a set of x}

Then Z should be a member of Z

But then Z cannot be a member of Z

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night and my fellow cousin ape.

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

Cardinality of a set is the measure of number of elements of the set

Cardinality of the integers is denoted with aleph-naught No