June
17, 2017 Saturday
Bedtime
Story
Analyzing the Predicate "Is a Prime"
As
I had said last night, now we will go for a more challenging example of a
natural number being a prime and see how it behaves within the Principia and
out of it.
We
shall begin with the arithmetical predicate ‘x is a prime’.
We
shall denote it with ‘pr (x)’.
This
as you will understand is merely an abbreviation of a simple English statement.
It
is not a string of meaningless symbols that would be found in Principia.
But
what would be its formal equivalent formula in Principia?
It
would be as follows:
‘~ (∃y) (∃z)
(x = ssy x ssz)’
It
is not too difficult to understand why and how this represents a prime number.
With
two successor functions in each of y and z, 0 and 1 are being knocked off.
Then
it says that x is NOT a product of any two natural numbers x and y that are
more than 2.
This
suffices to define a prime number though I agree that there can be more ways
than this to represent a prime number in the formal language of Principia.
We
can represent this formula of Principia by ‘Pr (x)’.
This
formula has a capital P.
Now
then, we want to make a false claim that ‘9 is a prime’.
In
formal language of Principia, it would be expressed as:
‘Pr
(sssssssss0)’
It
complete formula though would be (as agreed above):
‘~ (∃y) (∃z)
(sssssssss0 = ssy x ssz)’
The
quality of being a prime or the predicate ‘is a prime’ is primitive recursive
and hence yet again by virtue of Correspondence Lemma the string ‘~Pr
(sssssssss0)’ is a theorem of Principia.
Note
the titled sign in the beginning is a negation of 9 being a prime.
The
actual true formal formula of the above theorem would look like this:
‘~~ (∃y) (∃ z)
(sssssssss0 = ssy x ssz)’
Note
the double tilde signs preceding the rest of string of symbols.
So
let us go back to the primitive recursive predicate ‘dem’ and the statement:
“The
sequence of formulas with Gödel number x is a proof in Principia Mathematica of
the formula with Gödel number z”.
Here
too Correspondence Lemma will be invoked which will point out the fact that for
any true case of the number theory predicate dem (x, z), there should exist a
theorem which would like this:
‘Dem
(sss…sss0, sss…sss0)’ wherein the first strings of ‘s’’s is of length x and the
second string of ‘s’’s is of length z.
What
we have discussed here is critical for Gödel’s proofs.
We
will analyze this significance in the nights to come.
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.
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