June
24, 2017 Saturday
Bedtime
Story
The Fifth and the Final Point of
Today
we will briefly discuss the last and the fifth point of Gödel’s argument before
going into elaborate discussion of all the five.
(5)
Gödel finally constructed a formula A of the Principia that mirrored the
meta-mathematical statement:
‘Principia
Mathematica is consistent’.
We
will study this formula A and its construction in greater detail later on.
Then
he showed that the formula A
G can be derived within the Principia.
It
essentially means that if A is true then G is also true.
Remember
that formula G of Principia represents the meta-mathematical statement:
“The
formula G is not demonstrable using the rules of Principia”.
Gödel
demonstrated that the formula A cannot be derived within the axioms of the
Principia.
Since
the formula A could not be derived within the formal system, it logically
follows the Principia as a formal system is not consistent.
In
other words, not only the consistency of Principia cannot be established within
the system but furthermore the consistency is not reflected within the
system.
These
are the five main points around which the argument of Gödel was based and I
will devote my next few bedtime stories in developing them in greater detail.
One
thing I wish to clarify before that.
It
is of great relevance since it mattered a great deal to Gödel.
You
would have noticed all the while what I have been using the phrase “in the
Principia” all the time when describing the formal system.
This
perhaps might lead you to conclude, quite understandably, that these theorems
merely reflect the incompleteness and inconsistency of the Principia
Mathematica that was devised by Alfred North Whitehead and Bertrand Russell; One
specific formal system.
Gödel
suspected that the readers of his proofs who did manage to comprehend them
would fall for this trap and he was particularly concerned about this matter.
Hence
he made it a point to specifically point out that his results do not pertain to
merely one formal system but is a generality that concerns and pertains even
“to related systems”.
Let
me quote the two sentences that Gödel wrote down towards the end of his paper:
“Throughout
this work we have virtually confined ourselves to the system Principia
Mathematica, and have merely indicated the applications to other systems.
The
results will be stated and proved in fuller generality in a forthcoming
sequel.”
We
shall discuss in the nights to come about what he meant by these two slightly
enigmatic statements.
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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