May
10, 2017 Wednesday
Bedtime
Story
The Two Conclusions of Gödel's Incompleteness Theorems
Until
Gödel came into the picture, it was a generally accepted by almost all
mathematicians, great or trivial, that once a basic set of right axioms has
been chosen in a particular branch of mathematics, then all of the theorems
pertaining to it could be derived.
This
kind of surety was more so with number theory than with other branches of mathematics.
Geometry
was already considered conquered by Euclid except for that little glitch of
that parallel postulate.
Gödel
would come with his incompleteness theorems and spoil the party.
We
are now very close to the proof of Gödel’s theorems yet not completely near it.
There
is one more idea we will discuss and that is the idea of mapping in
mathematics.
Before
we do this, let me tell you what Gödel was successful in establishing.
Conclusion 1.
First,
he convincingly proved the impossibility of a meta-mathematical proof of
consistency of a system using the rules confined to the formal system such as
Principia Mathematica.
The
only way to prove the consistency of such a system would be to use
Transformation Rules and Rules of Inference above and beyond than those agreed
upon within the axiomatic system.
Hence,
even though the proof of consistency of such a formal system as Principia would
be achieved, it would come at a price of having had to resort to rules more
“superior” than the one agreed in the system.
This
itself would be a defeat in the face of a seemingly victory.
It
is much akin to the second labor of Heracles (popularly known as Hercules) who
was sent by Eurystheus to slay the Hydra of Lake Lerna.
Much
to the despair and anguish of Heracles, each time he would chop of the neck of
this hydra, this chthonic creature would display a botanical reaction,
sprouting two heads instead.
This
is exactly the condition when it comes to proving consistency of a formal
mathematical system.
This
relegates the proof from not being finitistic as Hilbert desired in his
program.
If
this was bad, Gödel came up with another conclusion that can only be described
as worse.
Conclusion 2.
He
showed that any mathematical system like Principia that is based on axioms is
essentially incomplete.
Try
to get the implication of this straight.
What
is says is that given any axiomatic system like Principia, which is a
consistent axiomatic system of number theory, there will be true statements or
theorems which cannot be derived from the system.
This
second conclusion is vividly visible in the arena of mathematics all over.
I
will take up one interesting example in the nights to come and show how it is
related to Gödel’s second conclusion.
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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