May
05, 2017 Friday
Bedtime
Story
So
now Gödel had devised a solid idea to check out the consistency of Principia
Mathematica.
That
is to show that there exists at least one formula (in those bizarre symbols)
that cannot be derived from the system using its own Formation Rules and Rules
of Inference.
Gödel
has invoked meta-mathematical reasoning upon the system.
He
then went on still further.
He
went on to seek a structural property of a formula such that it (it being the
property) fulfills these following three criteria:
[1]
The property is common to all the four axioms.
[2]
The property must be retained by the derived formulas and theorems
[3]
Not every formula and theorem that can be obtained using its rules must have
this property, or in other words there must be at least one formula that does
not have this property
If
these conditions are fulfilled, then too consistency of the system is established.
Let
me explain to you Gödel’s reasoning behind this.
This
is based on the agreed well-argued fact that in an inconsistent system, every
formula is a theorem.
In
a consistent system, there need to be at least one formula that is not a
theorem.
This
same reasoning is achieved if we design or conceive of a structural property
(much like a genetic trait) that can be passed from one axiom (ancestor) to the
next theorem (offspring).
Then
in the process if a derived theorem (suspected offspring) is discovered that
does not carry that structural property (genetic trait), then it cannot be
their descendent and hence not a theorem.
In
short, to prove the consistency of his formal system, Gödel had to devise a
structural property for his formal system and then derive a theorem that did
not possess this structural property.
I
hope you are not lost and still sticking on with Gödelian reasoning.
So
what kind of structural property did he seek out?
Let
us discover and explore a fascinating new word by the name of tautology.
For
me as a medical doctor, this is one of those strange words that sound familiar
or known but when asked to define, I would fumble and stumble and ultimately
get it totally wrong.
We
shall take up the story on this interesting word 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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