June
08, 2017 Thursday
Bedtime
Story
Gödel Numbering Series of Formulas
Now
we are very adept at assigning Gödel numbers to any one formula or a theorem.
The
question we were pondering last night was that having assigned Gödel number to
a series of formulas, how do we handle them?
Can
they cumulatively all be assigned one number?
Is
it possible to do that kind of feat involving a string of 10 or 20 such
formulas?
Yes,
it is possible using the exact same technique; generating a new number which
will be the product of first two primes, each raised to the power that
corresponds to the Gödel number.
So
if, say, we assign a number k to this series of two formulas, it will be
written as:
k
= 2m x 3n
This
same process can be repeated for as many formulas as needed.
All
this now should surely convince you that Gödel had devised such a system for
his proof that any expression of his formal calculus - may it be an elementary
sign, a string of signs comprising a formula or a sequence of such strings
representing sequence of formulas – all could be assigned a unique Gödel
number.
So
with his unique numbering system, Gödel was successful in establishing a
one-to-one correspondence of every expression of his formal system with a
distinct number.
Now
don’t get the wrong idea that given such a system, every natural number by
default will end up becoming a Gödel number.
(Just
consider the simple number 100, and think whether it can be a Gödel number or
not).
Yet
the fact remained that given any expression, it would be possible to figure out
its corresponding Gödel number which had to be special and unique.
Mathematicians
as I have told you earlier are never satisfied with what they have.
They
try to look at it in a reverse way or extend it further, anything to make your
and my life miserable.
So
let us try to think like them for a while and share their agony.
What
if you are given a Gödel number and asked to decode it and retrieve the
expression that it represents; can it be done?
Yes,
that too is possible.
In
our formal calculus that we have been considering, any number from 1 to 12
would represent the elementary signs.
But
what about Gödel numbers greater than 12?
How
can we decode them to arrive at their underlying formula?
We
shall be tackling this troublesome issue 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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