July
02, 2017 Sunday
Bedtime
Story
Arriving at the First Part of the Gödel’s Argument: If G is demonstrable, then ~G is demonstrable
I
agree that ω-inconsistency was a difficult sub-topic of mathematical logic and
it probably wasn’t easy to grasp the essence of it.
Yet
I was forced to write a little bit about it since Gödel had coined it, invented
it and used it in the proof of his theorems.
So
now let us go back to the point from which we had diverted to explain
ω-inconsistency.
Using
the stronger result obtained by John Barkley Rosser senior we landed up with
this formal antinomy or contradiction: G is demonstrable, if and only if, ~G is
demonstrable.
Yet
as had been agreed earlier, whenever a formula and its formal negation can both
be derived from and within the same system, then that arithmetical system is
not consistent.
As
a direct corollary of this, if the system of Principia Mathematica is
consistent, then within it, neither the formula G nor its negation can be
demonstrated.
Hence,
if the Principia is consistent, then G has to be formally undecidable.
Now
I will attempt to state the first part of the Gödel’s argument, namely, that if
G is demonstrable, then ~G is demonstrable, in a more formal manner.
The
Gödel way.
Let
us suppose that G is demonstrable in the formal calculus of Principia
Mathematica.
That
would imply that within the Principia there exists a series of formula that
serves as a proof for G.
We
had discussed long time back, when studying the basics of Gödel numbering, that
no many how many formulas there may be in a single proof, they can all be assigned
one unique and specific Gödel number.
So
let us assign the Gödel number k to this hypothetical series of formula that
serves as a proof for G.
We
also know that dem (x, z) is a number theory equivalent of the
meta-mathematical statement “such-and-such is a proof of so-and-so.”
In
that case the relationship dem (x, z) will hold true of x is k and z is
replaced with the value of Gödel number of G.
It
follows then that dem (k, sub (n, 17, n)) must be true and a fundamental truth
of arithmetic.
Now
assuming that dem (x, z) is a primitive recursive relationship (we cannot prove
each and everything in our bedtime stories), then its formal equivalent within
the Principia must be a theorem of the Principia.
The
formal counterpart within the Principia will appear like this:
Dem
(sss…sss0, Sub (sss…sss0, sss…sss0, sss…sss0))
The
number of ‘s’s in the above formula will be k, n, 17 and n respectively.
In
short, Dem (k, Sub (n, 17, n)) must be a theorem of the Principia.
Please
keep this thought in your mind.
We
will continue with this line of reasoning 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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