July
03, 2017 Monday
Bedtime
story
If the Formula G is Derivable, Then Its Formal Negation is Also Demonstrable
I
would request you to go back to the last night’s story before starting this
one.
There
is a rule of inference of the Principia that allows to derive from a theorem
type ‘P(k)’ (which means to say that the number k has the property P), the
theorem:
‘(∃x)
P (x)’.
This
theorem ‘(∃x)
P (x)’ implies that there exists some number x that was the property P.
Analogously,
using this very same rule of inference, from Dem (k, Sub (n, 17, n)) one can
derive the theorem:
‘(∃x)
Dem (x, Sub n, 17, n))’.
But
if you look back and see, this is a formal negation of G.
So
from having proved that Dem (k, Sub (n, 17, n)) must be a theorem of the
Principia, we have also derived the theorem ‘(∃x)
Dem (x, Sub n, 17, n))’.
Thus
it has been shown that if the formula G is derivable, then its formal negation
is also demonstrable.
From
this it follows that if the Principia is consistent, then G is not demonstrable
inside it.
On
similar lines it can be shown that if ~G is demonstrable, then the system of
Principia is ω-inconsistent.
But
being a long and a tedious derivation, I am going to give it a skip but if you
wish you can give it a try.
With
this, we have finished the second argument of Gödel and now we will move on to
elaborate on the third point.
(3)
Now many may not be very impressed with the conclusion of the second point.
By
many, I am referring to that extremely narrow bandwidth of human apes called
mathematicians who specialize in formal logic.
An
average ape like me may not be impressed or excited with anything that concerns
mathematics.
What,
they may ask, is so fascinating about constructing a formula within the
Principia that is undecidable?
Here
is what that is so sensational about the result that we obtained in point 2.
The
fascinating part that is so striking is that even though the formula G is not
decidable within the formal system of the Principia, yet it can certainly be
proved that its meta-mathematical reasoning is true.
Moreover,
since G and its formal negation ~G are making opposite claims about numbers or
anything to do with them, then one of them has to be true and the other has to
be false.
The
only question that should remain is which one is true and which one false.
In
fact, if you study the formula G and its negation carefully, it will become
obvious that it must be G which has to be true.
Let
us go back and see what the formula G stood for at the meta-mathematical level.
Its
meta-mathematical interpretation was, ‘G cannot be demonstrated within the
formal system of Principia Mathematica.’
Please
keep this idea in your mind as we will be carrying it 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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