August
09, 2017 Wednesday
Bedtime
Story
Undefinability Theorem vs. Incompleteness Theorems
As
I said last night, applying the diagonal lemma to this theorem gives a result
that is contrary to the equivalence stated below:
Then
for all A, the Tarski T-sentence True(g(A)) ↔
It
gives a result somewhat similar to the Liar’s paradox in a form of sentence S
such that S ↔ ¬True(g(S)) holds.
This
is an absurdity that contradicts the assigned Gödel numbers to its sentences.
I
hope you understand why this is an absurdity.
What
the result says is that the sentence S is equivalent to the negative of its
assigned Gödel number which patently cannot be true.
This
banality can even be understood by an ape with no mathematical instinct or
inclination.
Hence
no L-formula True(n) can define T*.
For
you people who are now well versed with Gödel’s reasoning for his
incompleteness theorems, Tarski’s proofs should not be that difficult.
Of
course, in its formal language it would be since we average and less-gifted
apes (I personally have no gift) are not well versed in the semantics of the
formal language.
But
as far as reasoning is concerned, there is extreme similarity except for the
diagonalization part for invoking the diagonal lemma.
There
is one interesting difference in the working of Tarski’s proof with that of Gödel.
In
Tarski’s proof, even though it is assumed that every L-formula has a Gödel
number assigned to it, no actual coding is required as in the case of Gödel’s
theorems.
Tarski’s
work I think ought to be seen as complementing Gödel’s work; while Gödel’s work
was restricted to formal mathematical systems like the Principia, Tarski’
extended it to systems of formal languages.
These
formal languages must fulfill certain criteria, such that they ought to have
negation, they should be capable of being self-referential and that diagonal
lemma must be applicable to them.
Very
broadly speaking, the conclusion of Tarski’s work can be stated as follows: No
sufficiently powerful language is strongly-semantically-self-representational.
Finally,
please recall what I had said at the end of Gödel’s incompleteness theorems.
The
proof of incompleteness theorems does not proclaim that there is a lack of
meta-mathematical proof of the consistency of the Principia.
What
is claims is that such a meta-mathematical proof even though may exist will not
be mirrored within the Principia.
Same
is the case with Tarski’s undefinability theorem.
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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