June
11, 2017 Sunday
Bedtime
Story
The Typographical Relationship of Meta-mathematical Statements to one another mirrors the Arithmetical Relationship of corresponding Gödel numbers to One Another
You
saw last night how the waiting queue of patients in a hospital could be
arithmetized.
Gödel
thought of applying this idea to the meta-mathematical statements of Principia
Mathematica.
Just
look what this brilliant mind came up with.
Since
every expression of the Principia has been assigned a unique Gödel number, then
any meta-mathematical statement about the formal expressions and their
typographical relationship to one another can be interpreted as a statement
about the corresponding Gödel numbers and their arithmetical relationship to
one another.
This
perhaps is the most imaginative and masterly use of mapping ever used by any
mathematician in the known history of mathematics.
The
above statement is not easy to handle so let me break down the logic and
connectivity to make it simpler.
We
know that the formal system of the Principia has now been coded with Gödel
numbers.
And
if this is so, then all the Gödel numbers will be related to one another since
like life, they are all derived from the basic sub units of twelve constants and
nine variables (in case of life the basic sub units are even fewer).
In
that case, all the meta-mathematical statements about each formal expression
will also be related to each other.
It
then follows that the typographical relationship of meta-mathematical
statements with each other will mirror the relationships that the Gödel numbers
in the formal system will have with each other.
Gödel
with this remarkable insight had discovered a technique to accurately mirror
within the formal logic system all meta-mathematical statements that described
the structural properties of expressions contained in it.
This
inspiring idea conceived by Gödel would later give rise to typographical number
theory that deal with systems that are capable of referring to themselves.
Let
us try out with a real but a simple example from the Principia Mathematica and
see if Godel’s idea of correspondence works out as being claimed.
Consider
a simple formula ‘~ (0=0)’
This
formula is completely wrong as it is saying that zero equals zero is not true.
Now
let us make a true meta-mathematical statement about this formula.
‘The
initial symbol of the formula ‘~ (0=0) is a tilde’.
According
to Gödel’s idea, any true meta-mathematical statement should faithfully map
itself into number theoretical assertion of the formal logic.
So
how does our true meta-mathematical statement behave?
Let
us find out.
We
shall perhaps find it out not tonight but 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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