September
17, 2017 Sunday
Bedtime
Story
The Proof comes out as deus ex machine
Last
night we had posed this question - are the set of provable* numbers and the set
of true* numbers identical?
To
get a positive answer to this question would require a exhaustive examination
of one-to-one correspondence between each element (number) of one to the other.
On
the other hand, for the answer to be negative there need to be merely one
property that holds true for one set but not the other.
Tarski
says there is indeed such a property that can be shown and it is a rather surprising
finding.
Tarski
considers this property very surprising almost startling that he likens it to dues ex machina.
Now
this term ‘dues ex machina’ is a
Latin term which in its own turn has been borrowed from ancient Greece.
It
literally translates into ‘god from the machine’.
What
it actually means to signify is a situation wherein to an apparently unsolvable
problem there suddenly arises a solution out of the blue.
It
was originally used in Greek tragedies as a plot device and we will leave it at
that.
Let
us go back and see what is that one property that holds true for one set but
not the other set (set of provable* numbers and the set of true* numbers).
The
relationship among the numbers of sentences can be characterized in terms of
simple arithmetical operations such as addition, multiplication and so on.
We
know that these terms occur in the language of our number theory.
From
this it then follows that the set of provable* numbers can also be
characterized using these very same operations.
By
doing this, in a manner of speaking, we have translated the definition of
provability from the meta-language into object language.
So
while we could achieve such a translation for the notion of proof, it may not
be possible to achieve a similar feat for the notion of truth.
This
is so because if this were to be possible, then the object language so
generated would in a way be semantically universal which then would endow it
with the power to generate paradoxes like that of Liar.
Tarski
say that it can actually be shown that if the set of true* numbers could be
defined in the language of arithmetic, then the Liar paradox could be
constructed in this object language.
The
only thing is that since the arithmetic language is a formal restricted one,
the paradox would not have the simple look that it originally has.
Now
since the set of provable* numbers is definable in the language of arithmetic
while the set of true* numbers is not is good enough reason to conclude that
these two do not coincide.
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.
Advertisements
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:
No comments:
Post a Comment