May
24, 2017 Wednesday
Bedtime
Story
Now
comes the second part of the Richard's paradox that brings in the concept of
mapping.
It
now asks us to map or link or associate each definition of integers that we
have so arranged (length-wise and lexicographically) with each natural number
starting from one.
So
the first shortest definition in alphabetical order will correspond to number
1, the next definition in the arrangement to number 2 and so on.
You
will naturally understand that the definition that has been assigned to each
natural number may not, or rather most likely will not, fit the natural number.
For
instance, number 8 might end up getting tagged with the definition ‘not
divisible by any integer except for 1 and itself’.
This
paired definition to number 8 certainly does not describe the property of 8.
And
in most instances it is bound to be so.
Yet,
in some rare instances the definition that has been assigned to a natural
number might actually end up in being the property of that number.
For
instance, if the natural number 17 gets tagged with the same definition
‘2-digit integer that is not divisible by any integer except for 1 and itself’,
then in this case the definition happens to describe this number unwittingly
but very suitably.
We
shall denote the former example when the integer is not paired with the
definition that describes it property as Richardian. (This is the whole crux of
the paradox)
Hence,
a natural number will be Richardian if the definition to which it has been
mapped with lacks its property.
So
we have to a state where defining ‘x is Richardian’ is just another way of
saying ‘x lacks the property that the defining expression of a serially ordered
set mapped to it suggests’.
As
you will understand, most, or nearly all the natural numbers will be Richardian
as it is more likely them to be paired with the definition that does not define
them.
At
least apparently, the property of being a Richardian is itself a numerical
property of integers.
This
implies that it should also be among the serially ordered definitions that were
created above.
Therefore,
the property of ‘x is Richardian’ must be assigned to some natural number.
Let
that number be n.
So
now if we ask the question, is n Richardian?
You
can obviously foresee a fatal paradox popping its dirty head out now.
If
you don’t, please go over the whole schemata once again.
We
shall go over once again 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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