July
12, 2017 Wednesday
Bedtime
Story
End of Gentzen and End of Incompleteness Theorems
I
had left you last night with the Prague Uprising aided by the incoming Russian
Liberation Army, a rebel faction of the Soviet Red Army.
Gentzen
was one of those Germans who were arrested during this Prague Uprising and then
handed over to the Soviets.
Soviets,
as you know, are well known for their lack of civility and crudeness.
So
in the Soviet prison camp, very soon after his arrest, on 4 August, 1945 Gentzen
died of malnutrition or perhaps even starvation.
Gentzen
in a 1936 paper had proved the consistency of Peano axioms and other formal
systems such as the Principia.
This
proof depends on arranging all the demonstrations of the Principia in a linear
order.
The
ordering is based on their “simplicity”.
This,
if you care to notice, has a ring of familiarity to it.
Yes,
it is very much based on the arrangement constructed in the Richard Paradox.
This
arrangement, as it turns out, bears a pattern of a certain “transfinite
ordinal” type.
If
you recall, we came across this term in our bedtime story series on Georg
Cantor.
There
is a rule of inference called “the principle of transfinite induction.”
The
proof of consistency is achieved by applying this rule to the linear order.
As
I had said earlier, this argument is not mirrored within the Principia.
The
other deficiency of this proof of Gentzen is that it is not finitistic.
You
will recall that David Hilbert laid strong emphasis on finitistic arithmetic
meaning that any consistency proof should use only finitistic reasoning about
finite mathematical objects.
Hence
the proof of Gentzen falls short of the objectives set by Hilbert in 1920.
With
this I have completed explaining the argument of Gödel concerning his
incompleteness theorems using as little formal logic and mathematics as
possible.
I
do not believe that any further simplification would be possible, not only by
me but by anyone.
To
round this up, I will briefly talk about the implications of Gödel’s incompleteness
theorems and what developments they spurred on.
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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