June
04, 2017 Sunday
Bedtime
Story
We
had ended last night with a statement that there exists an infinite class of
primitive recursive truths (or arithmetical truths) which if converted to the
formal language using the above table will end up in giving the theorems of
Principia Mathematica.
The
infinite class of primitive recursive truths includes something as banal as all
the additions, all the multiplications, statements such as “7 is a prime
number” or “20 is not a prime number”.
This
demonstration of high-fidelity correspondence between arithmetical truths and
interpreted theorems goes by the name of “Correspondence Lemma”.
I
want you to pay special attention to this concept of Correspondence Lemma as it
will be widely used in the proofs of the theorems later on.
This
Correspondence Lemma achieved several goals.
Firstly,
it confirmed that the Principia Mathematics is indeed a powerful axiomatic
formal system of number theory.
This
is shown by the fact that every primitive recursive truth end up in becoming
the theorem of Principia when encoded as a string of symbols governed by the
formal calculus.
Secondly
and more importantly, on a one-to-one basis, the intended interpretation or the
meanings assigned to the symbols is completely justified.
This
in essence is an exquisite link that intertwines truth and meaning.
Now
let us go back to the signs and symbols.
As
I had said earlier, the elementary signs of the formal logic are of two types,
constants and variables.
I
had enlisted 12 constant signs, their associated meanings and the associated
Gödel numberings.
Now
we will look at the variable signs.
Gödel
in his paper had designed three types of variable signs:
Numerical
variables
Sentential
variables
Predicate
variables
The
Numerical variables were assigned the alphabets x, y, x,…
The
numerical variables would represent either numbers or a numerical expression
such as x + y.
The
numerical variables are associated with specially designated Gödel numbers
which are unique prime numbers greater than 10.
We
shall study the three types of variables in the nights to come.
For
tonight, please think over and over the beautiful idea of Correspondence Lemma
and how it intertwines every primitive recursive truth into a theorem of
Principia when encoded as a string of symbols governed by the formal calculus.
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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