June
01, 2017 Thursday
Bedtime
Story
You
surely would have come across popular science books, but popular mathematics is
virtually an oxymoron.
That
is all I have to say about reading of mathematics and now I think you people
are optimally mentally geared up for the story on incompleteness theorems of
Gödel.
It
is not an easy task but I shall make my utmost endeavor to come clear on the
fundamentals behind these proofs.
Now
we are entering the haloed ground folks, the so called sanctum sanctorum of
Gödel’s work.
Most
apes around you will die ignorant of what this great mind logically deduced.
Those
are who are reading this are the lucky few to not only be aware of Gödel and
his work but if you take pain and effort, you might be the rare few who might
be even able to understand this remarkable feat in the history of formal logic
and computing.
We
will explore his work in three parts:
[1]
Gödel numbering
[2]
Arithmetization of meta-mathematics
[3]
Gödel’s argument
Gödel
numbering
Gödel’s
worked his theorems on a formal system that was largely an adaptation of
Principia Mathematica.
Hence,
much like the Principia, formal system of Gödel too had a class of elementary
signs, a set of primitive formulas or axioms, transformations rules and rules
of inferences from which further theorems were derivable.
In
fact so similar is his system, that for our discussion I shall be using the
word Principia or Principia Mathematica to refer to Gödel’s formal system.
We
have discussed formal systems in great length in our past bedtime stories.
This
was a formal coding of arithmetic to get rid of all its meaning.
Gödel
went further; he went on to code even this formal meaningless code (devised by
Alfred Whitehead and Bertrand Russell) with the simplest objects in the
universe – the natural numbers.
To
every symbol, well-formed formula and to each proof (finite sequence of
formulas) of his formal calculus, Gödel assigned a unique natural number.
Thus
each symbol, formula and proof is tagged with a unique Gödel number.
Why
did he do this?
It
was a technique, a method to determine the properties of any statement, such as
whether it is true or false.
Truthfulness
or falsity of a statement could now be determined by determining whether the
Gödel numbers associated with them had certain properties or not.
The
numbers would be long but what would matter was to show how those numbers were
constructed.
Do
not get discouraged if all this sounds gibberish.
Have
patience.
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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