June
23, 2017 Friday
Bedtime
Story
The Second, Third and the Fourth Point of
We
will continue where we left last night with the first point of Gödel’s argument
that pertained to the construction of the formula G.
Gödel
associated the formula G with its Gödel number g, and the construction is such
that it says:
‘The
formula that has a Gödel number g is not demonstrable.’
(2)
Further, as a part of his second argument, Gödel went on to show that the
formula G is demonstrable if and only if its formal negation ~G is
demonstrable.
If
you remember, this is once again very similar to Richard paradox where we came
to the conclusion that ‘n is Richardian if, and only if, n is non Richardian’.
Yet
we have agreed that in any formal system if both its formula and its negation
are demonstrable, then that system is inconsistent.
A
direct corollary of the above statement is that if the Principia is consistent,
then neither G nor ~G ought to be derivable from the axioms.
This
means to say that if Principia is consistent, then G is a formally undecidable
formula.
(3)
Gödel in his third argument showed that regardless of the fact that G is not
demonstrable, it still is a true arithmetical formula.
G
makes a statement that no integer can have a certain arithmetical property that
Gödel defined.
Since
Gödel proved that, it implies that G has to be true.
(4)
Once it is accepted that G is both true and formally undecidable within the
Principia, then it behooves us to accept the only logical conclusion; which is
that the Principia is incomplete.
Once
you accept that the Principia is incomplete, it stands to reason that all the
arithmetical truths cannot be derived from its axioms and using its rules.
Gödel
did not stop at that.
He
destroyed Hilbert’s dream by stating that not only was the Principia
incomplete, but it was essentially incomplete.
It
is reasonable if you are wondering about the difference between being
incomplete and being essentially incomplete.
By
being essentially incomplete, Gödel was emphasizing on the attribute that even
one were to bring into play ancillary axioms and rules into the Principia such
that formula G would then be derivable, then another true formula G’ could then
be constructed in the same manner which would remain undecidable and
underivable within that new reinforced system.
And
as you will understand, making further additions into the Principia to make the
new formula G’ derivable would similarly lead to yet another true formula G’’
that would be underivable within this twice reinvigorated system – and so on
and so forth.
This
is what is meant by the system being “essentially incomplete”.
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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