June
28, 2017 Wednesday
Bedtime
Story
Explanation Why Gödel number g of the Definite Formula G has to be sub (n, 17, n)
Last
night we were discussing the definite formula G of the Principia and that its Gödel
number g had to be sub (n, 17, n).
Here
I need to bring to your notice an important point.
There
is an important difference between the number and its mirror equivalent in the
Principia.
The
number is sub (n, 17, n) with a small s.
Its
equivalent in the Principia is the string of symbols that we had abbreviated to
‘Sub (n, 17, n)’ with a capital S.
In
simpler terms, ‘sub (n, 17, n)’ denotes an actual quantity in a manner similar
to the way ’10 X 10’ denotes the quantity of 100.
On
the other hand ‘Sub (n, 17, n)’ stands for meaningless strings of symbols of
Principia that signifies a number, very much like a number-naming string such
as:
‘ssssssssss0
X ssssssssss0’.
If
you look up, you will see that I had said that the Gödel number small g had to
be sub (n, 17, n).
Let
us try to see why this is so.
We
had admitted earlier that sub (n, 17, n) is the Gödel number of any formula
that emerges when the numeral n is substituted for the variable with Gödel number 17 (which is represented by
the variable y) inside the formula whose Gödel number is that very n.
Yet
the formula G was also obtained in exactly the same way.
While
trying to get the formula G, we had started out with the formula that had the Gödel
number n.
Then
all the copies of ‘y’ in it were replaced with the numeral n.
This
them makes it obvious that sub (n, 17, n) is the Gödel number of G.
We
also know that G is a formula within the Principia that mirrors the
meta-mathematical statement:
‘The
formula with Gödel number g is not demonstrable.’
That
implies that within the Principia, the formula G stands for the
meta-mathematical statement:
‘The
formula G is not demonstrable’.
So
what does this imply?
Simply
that the formula G of the Principia is stating that it is not a theorem of the
Principia.
This
is the end of the discussion of the first point of Gödel’s argument.
Now
we will move on to the elaboration of the second point of his argument.
I
am sure the first point will leave you sleepless the whole night if you are in
the Eastern hemisphere and keep you thinking the entire day if you are in the Western
hemisphere.
Of
course, if you are in the Middle East somewhere, try to keep yourself safe from
the religious madness and mass butchery of human apes.
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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