May
25, 2017 Thursday
Bedtime
Story
The Fatal Paradox:
Let
us go once back and understand the Richard’s Paradox in its other variant.
Let
us suppose that n is Richardian.
But
this is only possible if n is assigned a definition that does not define it.
This
means that n is not Richardian contradicting our assumption.
Now
let us suppose that n is not Richardian.
In
that case it will have its defining property ‘n is Richardian’ attached to it.
But
then by definition makes it a Richardian which goes against our assumption.
In
short, n is Richardian if, and only if, n is non Richardian.
This
makes the statement ‘n is Richardian’ both true and false.
Now
you may be wondering how on earth does this paradox of Richardian numbers fit
into our story of incompleteness theorems.
That
is an excellent question indeed.
Jules
Richard in framing his paradox had invented something that was never done
before: he had invented the idea of mapping meta-mathematical statements about
the formalized mathematical logical system to the arithmetical formulas within
the system.
If
you go back and consider his paradox, this idea of mapping was not the primary
intent; the mapping was being done primarily to generate a paradox or an
antinomy.
Let
us look at the example that we took previously to get this idea straight.
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.
You
would surely recall this statement in one of my bedtime stories not so long
ago.
The
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’ is a meta-mathematical
statement.
It
lies outside the arena of arithmetic.
The
numbers themselves namely the rational number 17.10101010… or 1693/99 are
themselves part of arithmetic which were linked to the above mentioned
meta-mathematical statement.
I
urge you to keep this in your memory as these ideas will gain ascendancy and
become prominent when we will go into the heart of Gödel’s theorems and logic.
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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