May
09, 2017 Tuesday
Bedtime
Story
Summarizing Gödel’s reasoning
Perhaps
because I went over Gödel’s reasoning over several nights that the thread may
have got lost or broken.
So
let go over once again over the entire chain of reasoning and summarize it
point wise:
[1]
Every axiom of our formal system is a tautology
[2]
Tautology behaves like a genetic trait that gets transferred to each derived
formula and theorem
[3]
Every formula that is derived from these axioms and hence any theorem of this
system is also a tautology
[4]
Hence any formula that is not a tautology is also not a theorem
[5]
One formula found that is not a tautology (in our case ‘v ∨
[6]
This formula then is not a theorem
[7]
Yet we had agreed that if this system of axioms were inconsistent, every
formula would be a theorem
[8]
Therefore the axioms are consistent
There
is another point that Gödel pondered over and I will mention in passing even
though it is not valid for our notion of absolute proof of consistency.
The
question that the logician asked is this:
Since
every theorem of this formal logic is a tautology and hence a logical truth,
does this also imply that any logical truth or tautology that is expressed in
the symbols of this system would be its theorem?
The
logician once again proved it to be yes, that is the case.
I
am avoiding all the proofs as that would be extremely tedious and may not make
for such cool bedtime stories.
If
it is so, (that all the tautological formulas of this system are theorems),
then the system would be deemed “complete”.
I
hope you have not forgotten that one of the requirements of Hilbert’s program
was completeness of the axiomatic system.
In
fact, not only Hilbert but every great mathematician right from the times of
Euclid has dreamt and expected this from the axiomatized system.
The
utopian hope was always to be able to generate complete and all exhaustive
theorems derived from a set of few basic axioms.
It
was the same with Euclid.
He
had hoped that from his few basic definitions and those 5 axioms, he would be
able to derive all the then known theorems of geometry along with all the
theorems that would be derived in future.
It
was to Euclid’s credit that he left his fifth parallel postulate an assumption
logically independent of the other four.
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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