July
28, 2017 Friday
Bedtime
Story
Liar Paradox and Gödel
To
resolve the Liar Paradox, Tarski had proposed a hierarchy of languages wherein
the key feature of the hierarchy would be that it would be permissible for
sentences in the higher semantic hierarchy to assign truth values to sentences
lower in the semantic hierarchy, but not the other way around.
Yet
such a system of “languages” has its own deficiency or limitation.
The
major limitation is that such a system of “languages” is incomplete.
Consider
the following statement regarding the system of “languages” that Tarski has
envisaged:
‘For
every statement in the level x of the hierarchy, there is statement at level x
+ 1 which asserts that the first statement is false.’
It
seems to be a fair, true and meaningful statement about the hierarchy that
Tarski talks about.
But
then it is applicable to statements at every level of hierarchy.
For
that to happen, it should be above and beyond every level of hierarchy which
only means that it cannot be within the hierarchy.
The
Liar Paradox is immensely intriguing and fascinating because not only it poses
a conundrum for the logicians but also has been exploited by them in proving
their unintuitive theorems.
Gödel
in his 1931 theorems had used a modified version of the Liar Paradox.
He
had replaced the sentence “this sentence is false” with “this sentence is not
provable” with that ingenious formula G.
In
a way, when Gödel explored the truth and the provability of his Formula G, he
was analyzing in a formal way the truth of the liar sentence.
I
shall end my discussion of the Liar Paradox with this much.
Now
I would like to move on to a very long paper that Tarski published in 1933,
very soon after Gödel’s 1931 landmark incompleteness papers.
In
this paper he attempted to give a mathematical definition of truth for
formalized languages.
In
1935 it was translated to German and it was only in 1956 that it managed to get
itself translated into English.
It
was all done by Tarski himself and he is the author of the book:
Logic,
Semantics and Metamathematics: Papers from 1923 to 1938.
The
book which is a collection of seventeen papers is considered to be a landmark
in the history of Analytical Philosophy.
In
case if you are wondering what the hell is this analytical philosophy, let us
read what Bertrand Russell had to say of it which was then a new and emerging
subject.
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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