August
03, 2017 Thursday
Bedtime
Story
Tarski's Undefinability Theorem
Though
the acclaim for the discovery of undefinability theorem is given to Tarski and
has been named after him, Gödel had discovered it independently way back in
1930 while working on his incompleteness theorems.
Obviously
Gödel never published it (if he done so it would now be named after him); though
he did write about it to the great John von Neumann in a letter in 1931.
Tarski
too was working on it somewhere around 1930 but in a paper that he had
presented to the Warsaw Academy of Sciences in March 21, 1931, he had only
reported some conjectures on it and not any proofs.
For
his proof Tarski was most likely inspired by the techniques that were invented
by Gödel such as Gödel numbering, coding, mapping and arithmetization of
meta-mathematics.
Much
like Gödel’s incompleteness theorems, Tarski’s proof is also an important
limiting result in mathematical logic, foundations of mathematics and in formal
semantics.
We
understand the idea of semantics in linguistics.
But
what is this formal semantics?
I
think by now you surely have gotten smart enough to associate the term “formal”
with mathematics and symbols.
Formal
semantics is a branch of both logic and linguistics that tries to understand
linguistic meaning using mathematical models.
The
theorem essentially says that in a sufficiently strong formal system truth
cannot be defined within the system.
Or
more informally, arithmetical truths cannot be defined in arithmetic.
In
his proof, Tarski showed that the procedures invented by Gödel for his theorems
are not applicable to semantic concepts such as truth (This is technical and I am
not in a position to explain the how of this statement).
Let
us see how Tarski stated his theorem bit more formally.
The
first version that I am offering to you now is not the exact Tarski’s version
but a simplified and a lesser formal version of it.
Still
it is something unfamiliar for most of us not acquainted with mathematical
logic.
Let
L be the language of first-order arithmetic or predicate calculus.
Remember,
Tarski’s work is only applicable to any strong formal language and not to our
natural languages.
Let
N be its standard structure.
Structure
of any formal language includes a set along with its finitary operations and
relations defined in it.
Relation
here refers to the property that assigns truth values to k-tuples.
Tuple,
as was once discussed earlier, is a finite ordered list of elements.
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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