January 10, 2017 Tuesday
Bedtime Story
John von Neumann Ordinal Construction
Let us see how von Neumann went about constructing the natural
numbers.
He very deftly created a successor function which is a primitive
recursive function in computer theory.
A successor function S is S(n) = n + 1 for each natural number n.
Thus S(1) = 2, S(50) =
51, S(100) = 101 and so on.
Von Neumann used this successor function for construction of sets
as follows:
Set 0 = {} which is an empty or a null set.
Define S(n) = n
{n} for every set n.
Thus with the successor function defined, it becomes automatic
that each set (of natural number) is equal to the set of all the natural
numbers lesser than it.
Then use the axiom of infinity.
This is how it works out:
{} = 0,
1 = 0 U {0} = {0} = {{ }},
2 = 1 U {1} = {0, 1} = {{ }, {{ }}},
3 = 2 U {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},
n = n-1 U {n-1} = (0, 1, 2…, n-1} =
{{ }, {{ }}…,{{ }, {{ }}…,}}, and so on.
This is the most elegant and beautiful way of both constructing
and defining the natural numbers based primarily on the ideas of Giuseppe
Peano, the Italian mathematician who made major contributions after 1890.
Giuseppe Peano like a true genius was born to nobody (a farmer in
a small village) of Cuneo in the Northern Italian region of Piedmont way back
in 1858.
You might be interested to know that this was also the time when
the word “girlfriend” first appeared in writing though till today there is no
proper or rigorous definition of what that word implies.
A region in Italy is equivalent to a state of the United States of
America of which there are twenty in the country.
Fortunately for Peano, mathematicians and for an average
storytelling chimpanzee like me, Peano’s genius was detected by his uncle when
he was 12 and he was dispatched to the University of Turin that was established
in 1404 and lies in the same region of Piedmont.
University of Turin is one of the oldest universities of Europe
and lies at the intersection of the beautiful Alps, coastal region of Liguria
and the wealthy region of Lombardy.
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.
The 20 Regions of Italy with Piedmont very nearly on top close to Alps, Liguria and Lombardy
Advertisements
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:
No comments:
Post a Comment