December 14, 2016 Wednesday

Bedtime Story 

Understanding the Axiom of Choice

Let me illustrate an example of choice function by creating a simple function.

Let there be an infinite number of boxes each containing infinite natural numbers.

Then I construct a choice function:

f(Box_n) = 10(n-1) + 1

In this case, this function will create a set by picking out one item from each box that will look like this:

{1, 11, 21, 31, 51…}

Axiom of choice goes one step further and generalizes this for all and any set, including infinite ones.   

This axiom is the most powerful of all and stands independent of all the other axioms of the set theory.

It has direct relevance to the question of continuum hypothesis put forth by Cantor.

The axiom of choice also leads to the well-ordering theorem.

The well-ordering theorem states that every set can be well-ordered.

If you recall, we had discussed the concept of well-ordered and total-ordered set some nights back.

A well-ordered set is a total-ordered set having a least element (classical example being the set of natural numbers).

Let me take you through few examples to get the relevance of this axiom and how it bears upon the continuum hypothesis. 

Take for example infinite number of bins containing pair of shoes one for left and the other for right foot.

This fact is the defining selection rule that distinguishes one shoe from the other.

So in this case one can pick out the right foot shoe from each bin without needing to invoke the axiom of choice.

Now suppose there are infinite such bins containing pair of socks that are actually meant for left and right foot but without any selection rule.

In this case, in order to pick out all the socks that are meant for the right foot, axiom of choice has to be inducted.

The axiom of choice demands that the socks are indexed as left and right making it possible to have an indexed family of socks (even if realistically we are aware that such a distinction does not work for socks).

Now you see mon ami, the axiom of choice does not seem all that obvious and logical and yet it has been taken as a fundamental assertion and assumption. 

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