May
23, 2017 Tuesday
Bedtime
Story
Analyzing Richard's Paradox and Its Variant (Richardian Numbers)
Richard’s
paradox is not so easy to grasp and so let us go over it once again.
Richard
to begin with, clearly defined an infinite list of English language sentences that
clearly express all real numbers.
In
the end, he created yet another sentence that describes a real number that HAS
to be different from any other real number defined before.
r,
to begin with, should have been one among rn.
Yet,
this new r is defined such that it has to differ from the other members of rn.
Doesn’t
it have a familiar ring to it?
Yes,
it is a virtual analog of the Cantor’s diagonal argument.
They
both are proofs of infinite sets or more accurately, uncountable sets that has
implications not only for our story on the incompleteness theorems but also on
Turing’s solution to the decision problem and Tarski’s undefinability theorem.
But
I have to say that stating Richard’s paradox this way does not make its
application for our story and thereby for Gödel’s theorems any meaningful.
For
it to have any bearing on our story, Richard’s paradox needs to be retold with
a slight variation.
This
variation of the paradox goes by the name of Richardian numbers and introduces
an important concept of mapping that was used by Gödel in an ingenious matter.
This
variation of the paradox uses integers instead of the real numbers but still
preserves the self-referential character of the original.
In
this case scenario, it begins with the assumption that arithmetical property of
each integer has been uniquely defined in whatever language one desires to
chose, which in my case is obviously English.
For
example, 1 can be defined as ‘the first natural number’.
2
can be defined as ‘the first even natural number’.
3
can be defined as ‘the first prime number’.
Just
like any system, even in this system not every property can be defined
explicitly as one has to start from somewhere with assumed primitive undefined
terms such as natural in natural numbers, integers, addition, products etc.
Hence
a definition can be ‘an integer is the product of two integers’ without us
having to define what an integer is.
Of
course, a prime number can be defined in terms of natural numbers but a natural
number would remain undefined and primitive.
So
now we have an infinite list of such definitions as the set of integers is an
infinite one (albeit a countable one).
Now
then as before, the paradox asks us to arrange these definitions first by the
number of letters in each of them and following that lexicographically.
We
shall continue with this variation of Richard’s paradox (Richardian numbers) in
the nights 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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