September
18, 2017 Monday
Bedtime
Story
The Conclusion
We
had came to the understanding last night that the set of these two numbers do
not coincide (the set of provable* numbers is definable in the language of
arithmetic while the set of true* numbers is not).
Then
it becomes conclusive that the set of provable* sentences and true* sentences
do not coincide either.
Going
a step further, using the definition of truth that we had discussed earlier, it
can easily be shown that all the axioms of arithmetic are true and all the
rules of proof are impeccable.
But
not all necessary have proofs.
Hence
all the provable sentences are true but the opposite does not hold good.
So
what do we end up with finally?
There
are sentences that are formulated in the object language of arithmetic that are
true but cannot be proved merely on the basis of axioms and rules of proof set
in the very beginning of arithmetical foundations.
Tarski,
exactly like Gödel, gives a parting warning on one special point regarding this
conclusion.
He
says that one might be tempted to think that this conclusion is dependent on
the specific axioms and rules of inference chosen for this arithmetic.
Perhaps
the conclusion may turn out to be contrary to this if we were to suitably add
on another set of axioms and rules of inference.
Unfortunately,
deeper study contradicts above notion.
The
conclusion would remain the same, i.e. all the true statements of a formalized
object language would not be provable within the context of object language.
This
would hold true for any formalized system, no matter how rich it would be in
terms of axioms or rules of inferences.
From
all this discussion it behooves us to once again reflect back on the Liar
Paradox and consider its role on human thought process.
At
the very beginning, Liar Paradox seemed kind of a recurrent evil that was set
out to destroy to smithereens any logic out there in the natural languages.
It
forced us to accept that in a natural language it would be all but impossible
to define precise notion of truth.
Since
logic and notion of truth seemed indefinable in a natural language, we tried to
restrict and limit its vocabulary
So
in the next step attempt was made to define the notion of truth in the more
restricted formal language which would be bereft of words of certain classes
such as demonstratives and deixis.
Even
that proved to be insufficient in preventing the generation of Liar Paradox.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
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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