December 15, 2016 Thursday

Bedtime Story 

Using Axiom of Choice for the Real Numbers

Let us take a more mathematical exam this time which may be more complex but is nearer to the implementation of the axiom of choice on the topic of continuum hypothesis.

Let S(i) be a family of sets that indexes real numbers.

Which means there is a set Si for each real number i.

Let them be:

S(1.2) = {1.2, 1.21, 1.201, 1.202…}

S(99) = {99.01, 99.02, 98.999999, 98.999999, 99.0,  99.00005…}

S( ) = {3.14159, 3.1416, pi, 3.14159265, 3.141593…}

S(1001) = {1001.00001, 1001, 1001.0002, 1001.003…}

S(0.0002) = {0.0002, 0.00019999, 0.000199999, 0.000200001…}

Each set that I have enlisted has non-zero and infinite number of elements.

S(i) = { S(1.2), S(99), S( ), S(1001), S(0.0002) }

The axiom of choice then allows to select a single element at random from each set forming a corresponding family of elements x(i).

So x(i) = { x_1.2, x₉₉, x , x₁₀₀₁, x_0.0002 }

The set x(i) is also indexed over the real numbers even though its elements were picked out or chosen from the collection of set S(i).

I am sure by now all of you (anyone crazy enough to have read all what I have written so far and given it any thought) would be lost.

Where did the axiom of choice start from and what it has anything to do with infinite sets.

The axiom of choice is an extremely simple assumption:

That when faced with infinite, a human mind or mathematics can choose or is allowed to choose from each set.

As simple as that.

But from this simple assumption, arises such powerful and counter intuitive results that can baffle any lay man and even great mathematicians.

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