March 31, 2017 Friday
Bedtime Story
Stating the Two Incompleteness Theorems
The paper had to be translated into English which itself proved to
be a hugely cumbersome task.
There were two reasons for it being so.
One for the difficulty and the novelty of the subject per se and secondly
and most importantly, for the personality of the author.
One of the translators was Jean van Heijenoort, a Frenchman of
Dutch ancestry.
He described Gödel as the most sedulously punctilious individual
he had ever known.
Between them they “exchanged a total of seventy letters and met
twice in Gödel’s office in order to resolve questions concerning subtleties in
the meanings and usage of German and English words.”
Not only did Gödel took his mathematical work very seriously, but
even the translation to English had to be meticulous.
Now let me state his theorems.
Gödel’s incompleteness theorems as is suggested by the plurality
of the noun, are not one but two.
Let me first state the two theorems in English (remember, the
paper was originally written in German).
First Incompleteness Theorem:
“Any consistent formal system F within which a certain amount of
elementary arithmetic can be carried out is incomplete; i.e., there are
statements of the language of F which can neither be proved nor disproved in
F.”
It must be brought to the notice that this theorem in the original
paper was published as “Theorem VI”.
The formal system that Gödel took into consideration was Principia
Mathematics of Whitehead and Russell.
Second Incompleteness Theorem:
“Assume F is a consistent formalized system which contains
elementary arithmetic. Then F ⊬
That strange symbol ⊬ between F and Cons (F) stands for “does not
prove”.
The term Cons (F) stands for consistency of F.
I will explain in detail what this statement means in detail.
This second theorem in the original 1931 paper was written as
“Theorem XI”.
Now having stated these theorems, let me go step by step what
these theorem imply.
First of all, these theorems restrict themselves to formal systems
of arithmetic.
Now what do we mean by a formal system of arithmetic?
We shall take up this formal system 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.
Advertisements
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:
No comments:
Post a Comment