June
09, 2017 Friday
Bedtime
Story
The Link Between
Tonight
let us try to figure out any Gödel number larger than 12.
Any
Gödel number greater than 12 will have to be decomposed into its prime factors.
This
can be done in one and only one way.
Why
so?
That
is one of the basic truths about numbers and hence it is known as the
fundamental theorem of arithmetic.
The
fundamental theorem of arithmetic states that every integer greater than 1
either is a prime or a product of prime numbers, and that this product is
unique, up to the order of the factors.
It
is possible that you may not be aware of it (I wasn’t) in spite of it being
centuries-old knowledge.
It
is found in Euclid’s Elements Book VII Proposition 31 as:
Any
composite number is measured by some prime number.
So
for example, consider the number 1500.
1500
= 2 x 2 x 3 x 5 x 5 x 5 = 22 x 31 x 53
So
if you see, first of all 1500 like any natural number greater than 1 can be
represented as a product of primes.
Secondly,
no matter how you break this up, the break up will always contain two 2s, one 3
and three 5s.
This
theorem is the principal reason why it was decided not to consider 1 as prime.
Making
one a prime would destroy the uniqueness of factorization as it is evident from
this simple case of 2 = 1 x 2 and 2 = 1 x 1 x 2 also.
So
you see mon ami, it was this fundamental theorem of arithmetic that Gödel was
exploiting for assigning those unique numbers.
As
I was saying, any number greater than 12 needs to be decomposed into its prime
factors and it will be unique as we have just discussed.
So
if the number is a prime number greater than 12 or such a prime raised to its second
or third power, then it must be a numerical variable, a sentential variable or
a predicate variable.
Furthermore,
if the number is a product of successive primes wherein each of them is raised
to certain power, then in that case that number would be representing either a
formula or sequence of formulas.
If
that is so, then its exact formula can be derived.
Let
us take a case example and see how that works.
Say
we are given a Gödel number 243,000,000 and it’s a question in logic paper of
your examination to decode it.
In
other words, what does the Gödel number 243,000,000 stand for?
We
shall take up this problem 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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