June
26, 2017 Monday
Bedtime
Story
Kurt
Last
night we were on this formula:
‘(∃x)
Dem (x, z)’
Just
to bring to your attention that the words affirmation or demonstrable refers to
formula being derivable within the Principia.
Adding
a tilde sign before this formula gives us its formal negation.
Then
we get this:
‘~(∃x)
Dem (x, z)’
I
am sure you can deduce what this formula states.
This
formula is a formal representation of the meta-mathematical statement:
‘The
formula with the Gödel number z is not demonstrable.’
Another
way of putting it would be:
‘No
proof can be shown for the formula with the Gödel number z.’
Gödel
in his theorems had shown that a specific case of the formula was not
demonstrable.
We
will see how he constructed this special case.
Consider
the formula below that we will label as Formula 1.
~(∃x)
Dem (x, Sub (y, 17, y))
This
formula has been taken straight out of Principia Mathematica.
Now
you will understand why so few apes (even mathematical apes) have ever had the
courage to read this book.
The
book is an unending series of - pages after pages and chapter after chapter -
such notations.
Yet,
since it is a formula of the Principia it is bound to have a meta-mathematical
statement that must correspond with it.
What
is that meta-mathematical statement?
You
will recall from the past bedtime stories that the expression ‘Sub (y, 17, y)’
represents a number.
It
needs to be pointed out immediately that even though the expression ‘Sub (y,
17, y)’ has been taken out from the Principia, it is not a formula but a kind
of function for identifying a number.
The
fact that ‘y’ is a variable and not a numeral, the expression ‘Sub (y, 17, y)’
cannot represent one specific number.
The
number that it represents is the Gödel number of the formula obtained from the
formula with Gödel number y by substituting for the variable with Gödel number
17 for the numeral for y.
In
that case the Formula 1 represents the meta-mathematical statement:
‘The
formula with the Gödel number sub (y, 17, y) is not demonstrable.’
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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