June
02, 2017 Friday
Bedtime
Story
Let
me elaborate on the idea of Gödel numbering along with the basic vocabulary of
our formal system which we have agreed to refer to as Principia Mathematica.
Let
us begin with elementary signs which form the fundamental vocabulary of the Principia.
As
we had agreed earlier, these elementary signs are of two types – constants and
variables.
Gödel
in his 1931 paper used just seven constant signs to set up his formal system.
It
is up to the creator of the formal system to decide and choose how many signs
he would like to have.
I
will show you a system that deploys twelve constant signs.
The
signs used here may be a bit different to that is currently in vogue (for
instance, the negation sign used here is the tilde but currently ¬ is in vogue),
but that is not very relevant as the idea is to get the principle behind the
entire proof.
Sign Meaning Assigned Gödel number
~ not 1
= equals 5
0 zero 6
s immediate successor
of 7
( punctuation mark 8
) punctuation mark 9
, punctuation mark 10
+ plus 11
x times 12
Many
of us non-mathematicians are not aware of the successor function s.
The
successor function is generally associated with either a number or a variable
which is a number and designates the natural number which would follow that
number.
The
symbol
Is the existential quantifier which simply
states “There exists”.
Some
of you might raise a question.
You
may question that the whole point of formal logic was constituted in the
Principia was to devoid the primitive symbols of any meaning.
Isn’t
this a pure case of double talk?
Well,
it seems so but there is a reason for it.
We
shall discuss this 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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