June
27, 2017 Tuesday
Bedtime
Story
Construction of the Formula 'G' with the meta-mathematical meaning
Last
night we were dealing with a formula that we had labeled as Formula 1.
~(∃x)
Dem (x, Sub (y, 17, y))
This
formula represents the meta-mathematical statement:
‘The
formula with the Gödel number sub (y, 17, y) is not demonstrable.’
Though
an alluring statement, it cannot a confirmatory statement since it involves a
variable ‘y’.
It
can only become a definite statement if y is replaced with one specific
numeral.
What
numeral can it be?
Let
us see what Gödel showed.
The
Formula 1 ~(∃x)
Dem (x, Sub (y, 17, y)), as already stated, is from the Principia.
So
it must have a Gödel number associated with it.
The
number would be huge and tediously long.
Yet
we need not worry and we need not take the pains of calculating it.
Let
us simply assign its value to be represented by the letter ‘n’.
Then
we will agree that whenever and wherever the variable ‘y’ appears, we will
replace it with the number n, or to be more accurate, with the numeral for the
number n.
We
shall represent this numeral for the number n as ‘n’.
(This
is much like writing ‘17’ when we know what we really mean to write is actually
sssssssssssssssss0.)
Doing
so will result in a new formula that will look like this:
~(∃x)
Dem (x, Sub (n, 17, n))
We
shall call this formula ‘G’.
Now
finally here we have the formula that is promising.
It
is based exactly on the Formula 1 but a very specialized form of it.
Its
meta-mathematical meaning is simple:
‘The
formula with the Gödel number sub (n, 17, n) is not demonstrable.’
Since
it has no variable, its meaning is definite.
Now
that this formula G is definite and occurs within the Principia, it must have a
Gödel number associated with it.
Let
us name the associated Gödel number as g.
Now
what can this number g be?
If
you study the lines carefully, you will come to the conclusion that the number
g has to be equal to:
g
= sub (n, 17, n)
If
you are not convinced about the number g, it is understandable.
We
shall discuss it over 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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