18
June, 2017 Sunday
Bedtime
Story
Perfecting a Distinctive Mapping System That Allowed Principia Mathematica to Accurately Talk About Itself
The
existence of the formula ‘Dem (x, z)’ or more accurately, ‘Dem (sss…sss0
,sss…sss0)’ inside the formal system of Principia shows or demonstrates that
true meta-mathematical assertions of the type “such-and-such demonstrates
so-and-so by the rules of Principia” are accurately mirrored in the theorems of
Principia.
Similarly,
every true meta-mathematical assertion of the type “such-and-such does NOT
demonstrate so-and-so by the rules of Principia” is also accurately mirrored in
the theorems of Principia.
These
two developments in the formal system are one of the fundamental contributions
by Gödel to logic.
With
this, Gödel had devised a distinctive mapping system that allowed the formal
system of Principia to accurately talk about itself.
The
system is accurately self–referential.
Now
there remains just one last bit to discuss about Gödel’s theorems before we
actually go into the meat of it.
Let
us see what it is.
Consider
once again the formula:
‘(∃x)
(x = sy)’
This
is one of the simplest formulas that uses the primitive recursive successor
predicate ‘s’.
If
you recall some nights back that we had assigned Gödel number to each of them,
but then to assign just one number to all those numbers, we had played a simple
trick.
We
had first 10 successive prime numbers raised each of them to these Gödel
numbers following which the product of these was assigned a number m.
So
we had this very large number substituted with m.
Now
you will also recall that ‘y’ in the formula is a numerical variable with assigned
Gödel number 17.
Now
let us try an idea.
What
if we replace the numerical variable y in the formula with the number m?
That
surely is permissible since y is a numerical variable and m represents the Gödel
number of the entire formula.
The
result of that will be one ghastly long formula that will look something like
this:
‘(∃x)
(x = sss…sss0)’, wherein the long chain of ‘s’s will be m + 1 in number.
Mind
you, the formula may be grotesquely long and extended but all that is says is
that there exists a natural number x that is an immediate successor of number
m.
In
other words, the number m has a successor.
Now
obviously this formula with a huge chain will have a gigantic Gödel number
associated with it (which is worked out same way as we did before by first
getting the associated numbers for each symbol and then raising them to the
successive primes followed by their product).
We
would agree it would be colossally huge and even writing it down would be an
arduous task; yet in principle it can be worked out.
Please
stay on to this chain of reasoning as we will continue 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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