July
07, 2017 Friday
Bedtime
Story
The Final Argument
We
will continue tonight with Gödel’s final argument where he is playing with the
number theoretical statement will be as follows, ‘There is at least one number
y such that no x whatsoever bears the relationship dem to y’ which in turn is
the mapping or reflection the meta-mathematical statement ‘There is at least
one formula of the Principia that is not demonstrable inside the Principia’.
So
then if we make the alterations as discussed above, then the meta-mathematical
statement ‘If the Principia is consistent, then it is incomplete’ can be
formally written down as:
(∃y)
~ (∃x)
Dem (x, y)
~ (∃x)
Dem (x, Sub (n, 17, n)
This
formula’s derivation will become clear if you go back to last night’s bedtime
story and ruminate over it for 10 minutes.
Yes,
that much time (and even more) is expected in understanding a formula of
mathematical logic.
This
formula is a part of the Principia.
This
large formula can further be simplified and shortened to:
A
⊃
G
(By
simple substitution of the left hand side of ⊃
Gödel
proved that this formula is derivable within the Principia.
Now
out of pure logic it can be shown that the formula that we designated as A is
not demonstrable within the Principia.
Let
us resort to a technique very widely used in mathematical proofs, which is
proof by contradiction.
Let
us assume that A is demonstrable within the Principia.
We
also know that the formula A ⊃
In
that case, by the Rule of Detachment, G too would be demonstrable.
But
we have shown that unless the Principia is inconsistent, G is formally
undecidable.
Which
means it is not demonstrable or derivable.
Thus
we have proved that if the Principia is consistent, then formula A is not
demonstrable in it.
So
what we ended up with?
To
begin with, formula A is a formal expression within the Principia of the
meta-mathematical statement or rather the claim that ‘Principia is consistent’.
Now,
even if outside the Principia by a chain of informal reasoning we could
establish this meta-mathematical statement, and then we could map each of
reasoning into sequence of formulas of the Principia, then the formula A would
be demonstrable within the Principia.
But
this as we have seen is impossible as long as the Principia is consistent.
So
we are left with one and only one conclusion.
If
Principia is consistent, its consistency cannot be established by any
meta-mathematical reasoning that has its formal counterpart within the
Principia!
All
I can say mon ami, that this logic of pure genius can come from no ordinary
mind and that if at least we average apes can understand it, we perhaps can
claim ourselves to be slightly bit more superior to our fellow cousin apes of
the larger family Hominidae.
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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