December 04, 2016 Sunday

Bedtime Story 


Continuum Hypothesis

Putting it in its more popular way, the continuum hypothesis states that there is no infinite set (let us say S) whose cardinality lies strictly between aleph naught  and c.

Symbolically, this can be stated as follows:

There is no set S (infinite of course), that would satisfy the condition given below

_{Aleph 0}< |S|< c

One can also state the continuum hypothesis in another form.

Remember how we had defined various cardinalities of infinities in the form of aleph naught, aleph one, aleph two and so on.

_{aleph 0}, aleph₁, aleph₂, aleph₃…

So if this is the case, then if aleph naught is the cardinality of the integers, then it follows from the continuum hypothesis that the cardinality of the real would be aleph one.

Symbolically,

 c = aleph 1

The continuum hypothesis leads to another very counter intuitive conclusion.

Every infinite subset of real numbers either has the same cardinality of that of the integers or same cardinality as that of the reals.   

Let me remind you that continuum hypothesis is only a hypothesis and not a proven theorem.

It was merely a belief (not a baseless one) of a radical thinker who had himself tried immensely hard but failed to prove it resulting in intense disquiet and restlessness.

Proving of the Continuum hypothesis was turned into a challenge at the International Congress of Mathematics of 1900 held in Paris.

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