May
22, 2017 Monday
Bedtime
Story
Understanding 'Lexicographical Order' and 'Canonical' in Richard's Paradox
We
will need to digress once again a wee bit to understand this new terminology
‘lexicographical order’.
The
word lexicographical is derived from the word lexicon that in turn is derived
from a similar Greek word meaning “of or for words”.
So
what exactly is a lexicon?
The
guys who study natural languages (as opposed to programming languages) are
called linguists.
According
to top linguists, our language essentially consist of two components, its
lexicon or the collection of words (its wordstock) and its grammar, the system
of rules that determines the rules of their combination to form meaningful
sentences.
Lexicographical
order is also referred to as dictionary order or alphabetical order.
It
came as a surprise to me that this terminology is as much a mathematical term
as much as it is used in linguistics.
In
mathematics, it is a terminology that belongs to set theory.
While
specifically it signifies arrangement in alphabetical order, in a more
generalized sense it pertains to the arrangement of elements in a finite
totally ordered sets.
So
if there exists a total order in the arrangement of elements of a finite set,
that would be called a lexicographical order.
Returning
back to our Richard’s paradox, when you arrange the sentences that precisely
define real numbers first length-wise and then lexicographically, the form of
arrangement that you get is called canonical.
Canonical
form is just a technical term in mathematics to state that something now has a
mathematical expression.
Canonical
is simple English refers to any object that has been sanctioned or authorized
by some supreme governing body.
So
now having achieved this canonical form of arrangement, we would have a list of
infinite sentences describing a specific real number arranged in a specific
manner.
To
get it straight, a definition that has lesser number of letters will precede
the one that has more number of letters.
Further,
if it happens that two definitions have the same number of letters, then
alphabetical order will determine their placement.
Having
done so, Jules Richard asks you to define a new real number r as follows:
The
integer part of r is zero, the nth decimal place of r is 1 if the nth decimal
place of rn is not 1, and the nth decimal place of r is 2 if the nth
decimal place of rn is 1.
Here
rn designates any possible real number
Please
note that in statement we are constructing a new real number r that is
different from any other rn in a very specific manner.
Now
all you have to do is glance back at the two statements of English language
that we created to define real numbers.
I
pray mon ami, go over this last part again and again and let it sink in.
We
shall go over it again to fully comprehend its construction and ramification.
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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