September
15, 2017 Thursday
Bedtime
Story
Formalized Number Theory in Semantically Restricted Language
The
vocabulary of this formalized number theory would consist of variables that you
are familiar with such as “x”, “y”, and so on, which would represent random
natural numbers.
It
would also have numerals “0”, “1”, “2” and so on that would represent specific
natural numbers.
I
hope you noticed the two different words used here, numerals and numbers.
We
had discussed it long time back again with respect to Gödel’s theorems.
The
vocabulary of number theory certainly would have symbols such as “+”, “-“, “X”,
and so on that would show relationship between numbers and operations that are
being performed on them.
Finally,
the vocabulary will contain logical terms that would either be sentential
connectives or quantifiers.
Sentential
connectives include these following four terms: “and”, “or”, “not”, “if”.
Quantifiers
are phrases in the form “for some number x”.
The
syntactical rules and the rule of proof will be kept to the simplest.
Now
I ask you mon ami to take your mind back to the first part of Tarski’s paper –
the notion of truth.
In
there we had touched upon the topic of object language and metalanguage.
We
had agreed then that from an object language, metalanguage can be constructed
and then from it adequate definition of truth could be formulated.
So
in this case of pure arithmetic with very limited vocabulary, we can in
principle use metalanguage to adequately define truths of the object language.
Say
for instance, we can claim that:
’50
X 2 = 100’ is true if two times fifty is one hundred.
It
is a valid metamathematical statement that adequately defines truth as we had
agreed upon in the first part of the paper.
This
way if we were to make a long list, it would not be wrong to say that we have
adequately defined set of true sentences.
After
all, the definition of truth says that certain specific condition that is
stated in the metalanguage is satisfied by all the elements of the set and only
by those elements.
In
our case, all the elements of the set are all true sentences.
So
now we are defining both truth and provability of our formalized arithmetic and
its limited vocabulary in a new metalanguage that can be termed
meta-arithmetic.
It
is only in this framework of meta-arithmetic that we will plan 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.
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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