May
17, 2017 Wednesday
Bedtime
Story
From Jules Richard to Georg Cantor
Cantor’s
diagonal argument is a mathematical milestone; something that he thought out
while trying to define sets.
This
proof or argument demonstrates that there exist infinite sets that cannot be
put into one-to-one correspondence with the infinite sets of natural numbers.
Not
only is the proof itself quite remarkable but the general technique that was
used by Cantor turned out to be a very powerful and an extremely useful one.
The
diagonal arguments after Cantor were later used in a wide range of seminal
works including Russell’s paradox, Richard’s paradox, Gödel’s incompleteness
theorems and Turin’s solution to the Decision Problem or as is famously known
in German language Entscheidungsproblem.
Some
day or rather some night I would like to tell you about the decision problem
too but when it will happen it is hard to say as that too is a decision
problem.
In
my previous bedtime that I had done on Cantor, I had stressed a lot on his set
theoretical notation and his continuum hypothesis but had bypassed the diagonal
argument.
Today
I am forced to return to it as it forms an invaluable part of our ongoing story
on Gödel’s incompleteness theorems.
While
the diagonal argument is always attributed to Cantor, there is another German
mathematician by the name of Paul du Bois-Reymond who had discovered it
independently round about the same time in 1875.
I
will merely outline the diagonal argument as published by Cantor in his 1892
paper without going into its implications.
The
proof of uncountable sets itself comes in two parts the diagonal argument of
which is just the first part.
An
Uncountable set or rather an uncountably infinite set is an infinite set that
contains too many elements to be countable.
Some
mathematicians replace the word countable with listable.
A
set of natural numbers is also an infinite one but it is considered a
countable.
This
holds true even for the integers which consist of all the natural numbers, all
the negative of natural numbers and the zero.
Though
apparently or intuitively, the set of integers seems to be double that of the
set of natural numbers, according to Cantor that is not so.
Based
on the principle of one-to-one correspondence between the elements of two sets,
the integers have the same cardinality as that of the naturals.
On
the other hand, Cantor showed that this one-to-one correspondence is not
applicable between the set of natural numbers and the real numbers.
Reals
or real numbers include naturals, integers, rational and the irrational numbers
(everything except the complex and imaginary numbers).
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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