June
30, 2017 Friday
Bedtime
Story
We
saw last night that Gödel also showed that if the formal negation of the
formula G was demonstrable, then G too would be demonstrable.
Thereby
we land up with this formal antinomy or contradiction: G is demonstrable, if
and only if, ~G is demonstrable.
I
will have to pause and add an important caveat here.
This
is not exactly what Gödel had proved in his original paper.
There
was an American logician by the name of John Barkley Rosser senior (1907 –
1989) who had done his doctoral research under the great Alonzo Church (most
famous for inventing the lambda-calculus).
In
1936, Barkley Rosser proved something that goes by the name of Rosser’s trick.
Now
what exactly is this Rosser’s trick?
Well…in
a simple way one can say that it is a proof of a stronger version of Gödel’s
incompleteness theorem.
Let
me explain how.
What
Gödel actually proved was that if G is demonstrable, then ~G is demonstrable
(rendering the Principia inconsistent) and if ~G is demonstrable, then PM is
ω-inconsistent.
So
Rosser’s trick is a method of proving Gödel’s theorem without the assumption that
the theory being considered is ω-inconsistent.
The
difference between Gödel and Rosser can essentially and informally be reduced
to these two statements:
While
Gödel used the formula that implies the liar paradox and says “This sentence is
not provable” in his proof, Rosser used the formula that says “If this sentence
is provable, there is a shorter proof of its negation.”
Since
Gödel used the term ω-inconsistency, we will need to understand what it means,
at least in simple terms.
Rather
than defining omega-inconsistency, let us understand ω-consistency.
It
was a theory that was introduced by Kurt Gödel during his work on the
incompleteness theorems.
It
refers to a collection of sentences that are not only syntactically consistent
(lacking any contradiction), but also avoids proving certain infinite
combination of sentences that are intuitively contradictory.
Let
us tackle it slightly more formally now.
This
formal consideration of ω-consistency will be carried forward to the nights to come in order to maintain the flow of thought.
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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