August
04, 2017 Friday
Bedtime
Story
Understanding Relation with Respect to Formal Language
Last
night I had left you with the structure of formal language that includes a set
along with its finitary operations and relations defined in it.
Relation
here refers to the property that assigns truth values to k-tuples.
De
Morgan was the first person who in his papers of 1860 defined relation that
bears the qualities very similar to those that are assigned to it today.
Augustus
De Morgan defined relation something like this.
When
two objects, qualities, classes or attributes, viewed together by mind, are
seen under some connexion, that connexion is called a relation.
Note
the spelling of connexion of the old English.
So
if L be the language and N its standard structure, then (L, N) is the
interpreted first-order language of arithmetic.
Let
x represent each sentence of this language L.
Then
each sentence x is assigned a Gödel number g(x).
As
I had mentioned earlier, Gödel while working on the proof of his theorems had
devised a bagful of tricks that would come in handy for the logicians that were
to follow his path.
T
would denote the set of L-sentences true in N.
T*
would donate the set of Gödel numbers of the sentences in T.
Tarski
then framed the question – Can T* be defined by a formula of first-order
arithmetic?
I
shall not go into the proof as that is not my primary goal and also due to the
fact that I have devoted plenty of time and effort in presenting to you Gödel’s
work which has left me quite exhausted mentally.
The
mathematical proofs are as it is very difficult per se, and to make it
interesting for lay readers only compounds the difficulties many fold
especially for an amateur writer like me who has almost no formal training in
serious mathematics.
Besides,
I do not have any flair or natural instinct either for higher mathematics or
for basic numbers.
All
that I have is a little bit of curiosity and there is a limit to what can be achieved
with just that no matter what Einstein may claim of himself.
Just
to let you know, Einstein is claimed to have said:
“I
have no special talent.
I
am only passionately curious.”
I
may accept a lot of wise stuff told by Einstein, but this is one thing I am
never going to buy.
Anyway,
let me take you back to Tarski’s undefinability theorem.
Tarski
came out with a proof that now goes by the name of Tarski’s undefinability
theorem.
It
is essentially an answer to this very question raised by him – Can T* be
defined by a formula of first-order arithmetic?
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
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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