May
21, 2017 Sunday
Bedtime
Story
From Cantor's Proof to Richard's Paradox
We
will continue with Cantor’s diagonal argument tonight.
I
hope you recall the first part of the proof wherein Cantor had first considered
a set T that presumably had in it all possible infinite (but countable) sequences
of binary digits.
Yet,
using the diagonal technique, he managed to create a sequence s that would
differ from all the enumerated elements in the set T.
Now
let us study the second part, which uses the common mathematical technique of
proof by contradiction.
We
had assumed in the very beginning that the set T contains in it all the
infinite sequences of binary digits.
Yet
by our (or rather Cantor’s) diagonal construct we proved that a new sequence
can always be created out of those infinite sequences of binary digits that
would be different from the ones contained in the set T.
Hence
the set T is uncountable.
Looks
childishly simple, right?
No
analysis, no complex numbers, no fancy theorems, and yet so many could not come
up with it before him.
Just
like Darwin and Wallace.
We
will leave it at that and return to Richard’s Paradox that was mentioned in
Gödel’s theorems.
Now
Richard’s paradox is a semantical antinomy that involves both set theory and
our natural language.
It
is not as straightforward as Cantor’s diagonal argument (in spite being
fundamentally based on it) and requires some mental effort.
It
is good to exercise your mental faculties once in a while; it is exhilarating.
So
let us go straight into it.
Jules
Richard first makes an observation that certain expressions of our natural
language can define real numbers very precisely while certain expressions
cannot.
This
statement needs clarification with examples.
The
following statement, ‘The real number the integer part of which is 1 and the
nth decimal place of which is 0 if n is even and 1 if n is odd’ unambiguously
defines the real number 17.10101010… which represents the fraction 1693/99.
Then
there are certain statements that do not define a real number, example being
“Most apes fear mathematics”.
We
need not worry about those statements that cannot define real numbers; neither
do we have to worry about some real numbers that would be hard to define using
English language.
So
there has to be an infinite number of such statements that define real numbers
precisely; their lengths do vary but they are of finite lengths.
Now
Jules Richard asks of us to arrange these statements first length wise and then
in lexicographical order.
I
request you to go over at night what Jules Richard asks of you.
He
is asking you to link our natural language with real numbers (mind you, not the
natural numbers, not the integers or the rational numbers but the ones that
contain 𝜋, 𝑒,
and √2.)
I
hope to continue on Richard’s Paradox in the nights to come more leisurely.
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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