June
15, 2017 Thursday
Bed
time Story
Going Deeper
By
now we understand the idea of arithmetization of meta-mathematics.
The
typographical properties of the long chain of symbols very accurately can
replace and reflect the properties of prime factorizations of very large Gödel
numbers.
Now
we will move on further and study other key concepts necessary to comprehend Gödel’s
logic.
Consider
the following meta-mathematical statement:
“The
sequence of formulas with Gödel number x is a proof in Principia Mathematica of
the formula with Gödel number z”.
This
statement speaks about typographical relationship between certain strings.
But
because meta-mathematics has been arithmetized, this statement is mirrored
inside number theory by a statement about a pure numerical relationship between
numbers x and z.
Let
this pure numerical relationship between numbers x and z be demonstrated by the
abbreviation ‘dem (x, z)’.
You
may wonder why the letters ‘dem’ were chosen as the abbreviation.
‘dem’
is a diminutive of the word ‘demonstration’, and hence will serve as a reminder
of the meta-mathematical relationship to
which this pure number theoretical relationship corresponds to.
That
meta-mathematical relationship is this:
‘The
sequence of formulas with Gödel number x is a proof – and therefore a
demonstration – inside Principia Mathematica of the formula with Gödel number
z.
It
is assumed, of course, that the numerical relationship that ‘dem’ denotes or
will denote is dependent on the axioms and rules of inference of the formal
system.
Any
change made in the formal system or the Principia would alter the numerical
relationship that ‘dem’ will denote.
Gödel
in his paper devoted a great part and went to great length to convince the
readers that dem (x, z) is a primitive recursive relationship.
If
this is accepted (which we are going to), then from the Correspondence Lemma it
automatically follows that there has to be a formula within the Principia that
expresses this relationship but only in the formal notation.
Let
us denote this formula with Dem (x, z) with a capital ‘D”.
Another
point that needs to be emphasized here is as follows:
Consider
two numbers 2 and 5.
Then
dem (2, 5) would be meaningful statement about the natural numbers 2 and 5, yet
it would be clearly false as 2 is not a Gödel number of any proof and 5 is not
a Godel number of a complete formula.
We
know this because 2 was assigned to represent ‘
‘
(‘or’) and 5 was assigned to represent ‘=’ (equals).
Now
consider its formal counterpart ‘Dem (ss0, sssss0)’.
This
is merely a string of symbols from Principia and hence it is meaningless.
So
it is neither true nor false.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
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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