August
10, 2017 Thursday
Bedtime
Story
What Incompleteness and Undefinability Means
One
must be careful in interpreting the significance of both Tarski’s and Gödel’s
theorems.
Just
because a truth can’t be defined in one theory does not exclude its being
defined in a higher and stronger theory.
For
example, the set of formulas of the first-order Peano arithmetic that are true
in N (set of natural numbers) may not be definable in the first-order
arithmetic.
But
they surely are definable by a formula in the second-order arithmetic.
Further
if we go, the set of true formulas of the second-order arithmetic can be
defined by a formula in first order Zermelo-Fraenkel set theory or ZFC.
That
C stands for that famous axiom of choice on which I had spent so much time some
time back.
And
yet again, the consistency of the ZFC theory cannot be proved within the theory
itself.
With
this we have finished the third major work that Tarski had done on logic.
Now
I shall describe Tarski’s 1969 paper “Truth and Proof” wherein he takes into
consideration both Gödel’s incompleteness theorems and his undefinability
theorems and their implications for the axiomatic system of mathematics.
Tarski
himself does not rank this paper as original.
He
raises the problem of Liar’s Paradox and how it raises the concern of truth in
natural languages.
Tarski
divided the paper into three parts and so in respect to him, so shall I.
The
three parts are as follows:
[1]
The Notion of Truth
[2]
The Notion of Proof
[3]
The Relationship of Truth and Proof
Let
us begin with the notion of the truth.
There
is a broad way of considering the term “true”.
From
a psychological point of view, the truth can have emotional aspect to it.
Most
human apes consider their beliefs, both emotional and religious, to be true
even though they directly contradict each other and reality.
Then
there is that vague “inner truth” to which the spiritualists and the other
usual suspects refer to that is beyond any definition or realm of
investigation.
Tarski
makes very clear that his definition of “true” totally eliminates the above
types.
The
truth that Tarski is dealing with lies in the realm of logic and hence the
usage of “true” in this paper is very specific and restricted.
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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