June
29, 2017 Thursday
Bedtime
Story
How Gödel Proved That G is Demonstrable Only if its Formal Negation ~G is Demonstrable
(2)
In the second point, we will try to make sense on how Gödel proved that G is
demonstrable only if its formal negation ~G is demonstrable.
This
argument bears extremely close semblance to the concept of Richard’s paradox
and Richardian number though enshrined with subtle but critical differences.
At
the very onset, let me point out the similarities and differences between Gödel’s
argument and Richard’s Paradox.
The
crucial point that has to be specified is that G is not exactly same as the
meta-mathematical statement that it mirrors, but it merely represents it within
the Principia.
In
the Richard paradox of you recall, the number n is associated with a certain
meta-mathematical statement.
In
the case of Gödel, the number n is associated with a certain formula that
belongs to the Principia, and this formula more by coincidence than intent
happens to represent a meta-mathematical statement.
In
the concept of Richard’s Paradox, the question that was eventually raised at
the end was whether the number n has in it the meta-mathematical property of being
Richardian.
In
Gödel’s case, the question that was raised is whether the number g = sub (n,
17, n) carries a specific arithmetical property, that property being the
assertion ‘dem (x, g)’ holds for no cardinal number x, whatever the number x
may be.
What
I am trying to say was although Gödel constructed his argument based on
Richard’s paradox, he kept the distinction very clear between the statements
within the Principia and the statements about Principia.
Hence
it is totally free from any sort of fallacy that many consider exists in
Richard Paradox.
Gödel
started this part of his proof by showing that if the formula G were to be
demonstrable in the formal system, then its formal negation would also be
demonstrable.
You
would recall what the formula G is.
~(∃x)
Dem (x, Sub (n, 17, n))
It stands for the meta-mathematical statement:
‘The
formula with the Gödel number sub (n, 17, n) is not demonstrable.’
So
then, the formal negation of the formula G would be:
‘(∃x)
Dem (x, Sub (n, 17, n))’
A
negation of negation is obtained by simply taking off the tilde sign (negation)
from the formula G.
The
meta-mathematical interpretation of this formula would be:
‘There
exists a demonstration of the formula G within the Principia.’
Further
on, Gödel also showed that if the formal negation of the formula G was
demonstrable, then G too would be demonstrable.
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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:
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recommend this large collection of Halloween Songs for Kids:
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