June
14, 2017 Wednesday
Bedtime
Story
The Proof That Principia Mathematica ('and related formal systems') are Self-Referring System that in a Way Deeply Connects Truth and Meaning
As
I was saying before I broke off and went to sleep, ‘x is a factor of y’ is an
arithmetical predicate.
A
predicate is a statement that can be true or false depending upon the value of
its variables.
Besides
that, it is also primitive recursive.
(Most
functions that are commonly studied in number theory are primitive recursive.
For
example, addition and division, the factorial and exponential functions are all
primitive recursive functions.)
Another
example of a primitive recursive function is the successor function (which
returns the immediate successor of its argument) that we have been using.
This
then allows us to invoke Correspondence Lemma for the predicate ‘x is a factor
of y’.
Then
it follows that the string of symbols that define the predicate ‘x is a factor
of y’ is a theorem of Principia Mathematica.
So
what have we ended up with?
That
there is a theorem of the formal system (Principia Mathematica) which is a
translation of a true meta-mathematical statement “The initial symbol of ‘~
(0=0)’ is a tilde”.
So
finally here in Principia Mathematica we have a self-referring system in the
sense that truthful meta-mathematical statements about mathematics are
reflected in the theorems contained in it.
This
is achieved using two ingenious concepts:
[1]
Gödel numbering and the idea of faithful mapping
[2]
Correspondence Lemma: which essentially confirms two facts about a formal
system:
[a]
Every primitive recursive truth, when encoded as a string of symbols of the
formal calculus, is a theorem and
[b]
On a one-to-one basis, the formal symbols merit their intended interpretations.
In
a manner of speaking, truth and meaning are deeply connected.
The
discovery that typographical properties of the long chain of symbols very accurately
can replace and reflect the properties of prime factorizations of very large Gödel
numbers is known as arithmetization of meta-mathematics.
If
you think this was difficult, then things are going to get even tougher.
So
I shall stop over here and would like to think back all that we have discussed so
far.
Today
the conclusion that we have reached about the Principia is of vital importance
and has far reaching consequences.
It
is a self-referring system that contains within it theorems that mirror true
meta-mathematical statements.
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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