September 24, 2017 Sunday
Bedtime Story
Comparing
Gödel’s completeness theorem says that in a
system of first-order logic, there are sentences or axioms that are consistent,
and if a sentence in it is provable using these axioms and rules of inference,
then it will be true in all models of theory.
Gödel’s incompleteness theorem on the other
hand says that given a consistent, computable set of axioms there is a
statement φ in the language of arithmetic such that the axioms prove neither φ
nor ¬φ.
One must not confuse “true” with
“derivable” (or provable).
So the completeness theorem says that given
a set of first-order logic axioms in the language of arithmetic, there is a
model in which these axioms and their consequences are true.
It does not follow from this that these
axioms will allow or are sufficient to prove a arithmetical sentence or
disprove it.
Thus there remains a chiasm between what is
true and what is provable in a first-order logic system of arithmetic.
It was problems such as these that was
bothering David Hilbert and for which he had proposed the Entscheidungsproblem
regarding validity of a statement of first-order logic to all models.
Is there way of knowing this answer?
Is there any such procedure?
Is there any such algorithm?
His problem was posed in 1928.
He did not have to wait very long (just
eight years) as the answer came in 1936 and not from one but from two
independent separate sources by two men, each answering in their very original
way.
They were Alan Turing who answered this
question with the concept of his Turing Machine and Alonzo Church who solved
this problem with his lambda calculus.
Alan Turing has been highly popularized
both in books and movies and many if not most modern humans seem to be faintly
aware of him and his work, most particularly for his code-breaking work at the
Benchley Park.
Alonzo Church who happens to be both an
American and doctoral advisor of Alan Turing is far less known in popular media
and to the general public.
A closer inspection of the private life of Alonzo
Church made me reconsider something that I once held as a long cherished belief.
It is the connection or rather disconnect
that ought to exist between reason/logic and religion.
There is a school of thought which reasons
out that the brain of human apes is inherently illogical as is evident from the
wide-spread fear of mathematics prevalent in them along with almost pervasive belief
in mutually contradictory religions/gods and spiritual mumbo jumbo across the
cultures.
The argument is that mathematics is pure
logic and since most of us find mathematics not only difficult but even scary
should indicate that human brain is anything but logical.
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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