July
06, 2017 Thursday
Bedtime
Story
The Pièce de Resistance of Gödel’s Argument
(5)
Now we are on to the last leg of Gödel’s argument, the grand finale and
postlude, the pièce de resistance of Gödel’s tour de force.
The
argument in this final part is based around the meta-mathematical statement,
“If Principia is consistent, it is incomplete”.
This
conditional statement when taken as a whole, it can be shown, is represented by
a demonstrable formula within the Principia.
That
demonstrable formula was constructed before.
Many
nights ago, and you may have to search out that night’s bedtime story, we had
agreed that the meta-mathematical statement ‘Principia Mathematica is
consistent’ is equivalent to the statement ‘There is at least one formula of
the Principia that is not demonstrable inside the Principia’.
In
fact, that night’s story was told precisely in the event that I remain alive to
tell tonight’s story.
Now
let us map the meta-mathematical statement ‘There is at least one formula of
the Principia that is not demonstrable inside the Principia’ into its
number-theoretical equivalent.
That
number theoretical statement will be as follows, ‘There is at least one number
y such that no x whatsoever bears the relationship dem to y’.
This
statement can be further reframed as ‘Some number y has the property that for
no x does the relationship
dem
(x, y) hold.
This
in the formal language of Principia would become the following equation:
(∃y)
~ (
x)
Dem (x, y)
Let
us label this formula of Principia as (A).
There
is yet another meta-mathematical statement that can express this formula of
Principia.
‘There
is at least one formula [whose Gödel number is y] for which no proposed
sequence of formulas [whose Gödel number is x] constitutes a proof inside the
Principia.’
This
formula that we have labeled as A forms the preceding condition of the
meta-mathematical statement, ‘If the Principia is consistent, then it is
incomplete.’
The
second part of the statement ‘it (the Principia) is incomplete’ is tantamount
to the statement ‘x is not a theorem of the Principia’, wherein x is a true
formula that is non-demonstrable.
Such
a formula exists and that is the formula G.
In
that case, we can write the second part of the statement by saying ‘G is not a
theorem of Principia.’
If
we can convert this statement into meaningless chain of strings, then it would
be a part of the Principia.
That
‘G is not a theorem of Principia’ is a claim made by the formula G itself.
It
then follows that we can replace the second part of the statement with G.
Hold
on to this line of thought and logic while I take break for the night.
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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