January 06, 2016 Friday
Bedtime Story
Finishing off the Banach Tarski Theorem (Sorry if I let you more confused than ever before)
The poles of the sphere in the Banach Tarski theorem can be
treated like the rooms of the Hilbert’s Paradox of Grand hotel.
Or they can be treated like a hole in a circle when we discussed
the concept of equidecomposablity.
The poles and the center can be similarly filled by shifting the
points on the circle slightly to the left and that will be it.
This is an extremely crude and nonmathematical way of
understanding this famous Banach Tarski paradox.
It is important to recall that Banach Tarski paradox is a direct
outcome of the axiom of choice, the 6th axiom of the Zermelo Set
Theory.
The irony is that these axioms were proposed to get rid of
paradoxes and antinomies that the naïve set theory of Cantor gave rise to.
Now we can proceed to the last and the 7th axiom of the
Zermelo set theory.
7. Axiom of Infinity:
It is rather surprising but it is true that the other 6 axioms
enlisted before are insufficient to prove the existence of the set of all
natural numbers.
So an axiom was created by Zermelo to state the existence of the
set of natural numbers.
Technically, the axiom goes as follows:
There is a set I (infinite), such that the empty sets is in I and
such that whenever any x is a member of I, the set formed by the union of x
with its singleton {x} is also a member of I.
In simple words all it says is that there is a set, I, that
includes all the natural numbers.
The great John von Neumann had indirectly conceived of this axiom
for formulating a technique for the construction of natural numbers.
We shall see how von Neumann went about constructing the natural
numbers in the night to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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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:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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