August
05, 2017 Saturday
Bedtime
Story
The Statement of
Tarski
answered his own question by developing a theorem (and hence a rigorous proof)
that now goes by the name of Tarski’s undefinability theorem.
The
theorem is very simple to state and goes something like this: There is no
L-formula True(n) that defines T*.
That
odd word True(n) is the truth predicate.
It
is one of the basic and fundamental concepts of a formal language.
It
is the formal way of expressing that a sentence is true.
Another
way to state the theorem would be as follows:
There
is no L-formula True(n) such that for every L-formula A, True (g(A)) ↔
In
casual language, what this means that in a formal arithmetical system, the
concept of truth is not definable if one has to stick to the means that
arithmetic provides.
This
in way, not only reflects the limitations of any strong formal arithmetic
system, but a major limitation to the idea of self-representation.
This
means to say that it is impossible to define a formula True(n) whose extension
is T*.
The
only way it would be possible to do so would be to invoke meta-language.
This
is so because the expressive power of the meta-language always exceeds that of
the formal language L of first-order arithmetic.
The
contents of metalanguage always exceed that of object language concerning
axioms, primitive notions and rules of inference.
Metalanguage
will always contain in it more provable theorems than will the object language.
In
short, a truth predicate for the first-order arithmetic can be defined in
second-order arithmetic.
You
would recall that we had discussed first-order arithmetic when we were into the
history of mathematical notations and were talking about the Italian
mathematician Giuseppe Peano and his axioms.
Those
were the most fundamental axioms for generating the natural numbers and till
date they have remained so.
Later,
David Hilbert and Paul Bernays (yes, the Swiss mathematician who was expelled
from Göttingen for being “Non-Aryan”) in their book Grundlagen der Matematik
(Foundations of Mathematics 2 volumes 1934 and 1939), introduced second-order
arithmetic.
Second-order
arithmetic formalizes natural numbers and sets of natural numbers.
Further,
indirectly by using coding techniques, it can also formalize other mathematical
objects such as integers, rational numbers and real numbers.
This
was noted later by Hermann Weyl, who had been associated with both the world’s
greatest centers of mathematics, University of Göttingen and IAS, Princeton.
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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