September
16, 2017 Saturday
Bedtime
Story
Attacking the Problem: Whether the Set of Provable Sentences are same as the Set of True Sentences
Last
night we had stopped at the junction of having made framework of
meta-arithmetic in which the plan was made to study and analyze the problem
that we posed in the beginning of this third part – namely, whether the set of
provable sentences are same as the set of true sentences.
Tarski
does not go into the detail of his proof but says the solution of this problem
turns out to be a negative one.
Which
means to say that the set of provable sentences do not correspond always with
the set of true sentences.
As
you can very well guess, this proof is intimately related to that of the famous
proof of Gödel that he deployed for his incompleteness theorems.
I
will try to explain to you how this was proven.
The
reasoning should not be that difficult for you guys now that you have been
through Gödel’s proof.
We
had agreed that metalanguage is always richer than the object language, as it
contains the entire object language as a part of it besides the terms needed to
discuss the object language.
Metalanguage
will contain within it expressions of sentences, of sets of sentences, of
relations among sentences that will allow to study the properties of different
objects and the relationship between them.
Now
what will be done next should remind you something we discussed in past.
I
shall reveal it you; it is the Richard’s Paradox.
Like
then, here too all the sentences of the object languages are to be arranged in
an infinite series from simple to more complex.
Then
every such sentence will be matched with a unique natural number such that any
two numbers that are matched with any two different sentences are not same.
By
this, a one-to-one correspondence is established between the sentences and
numbers.
Once
this is established, other such correspondences can be established such as
between sets of sentences and sets of numbers or say between relations among
sentences and relations among numbers.
If
this is possible, then we can also think about making one-to-one correspondences
between numbers of provable sentences and numbers of true sentences.
Tarski
assigns a simple sign to these provable numbers and true numbers.
We
will assign an asterisk mark to them and call them provable* numbers and true*
numbers.
Having
done this we can reduce our original question which was vague and general to
this more precise and pointed one: are the set of provable* numbers and the set
of true* numbers identical?
To
get a positive answer to this question may be a difficult task as we need to
make sure there is a complete one-to-one correspondence between each element
(number) of one to the other.
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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