August
06, 2017 Sunday
Bedtime
Story
Second-Order Arithmetic
I
want you to get a general feel of the second-order arithmetic so I will briefly
describe its workings.
Firstly,
the language of second-order arithmetic is two-sorted, meaning it studies the
world in a pattern of two.
It
is somewhat like a biologist dividing living creatures of the planet into
plants or animals or more universally, into non-cellular life (viruses and
viroids) and cellular life.
I
was not aware, but the viroids are the simplest known life forms consisting
only of short (as short as 359 nucleotides) string of circular single-stranded
RNA without even having the protection of any protein or fatty coat.
Just
a bare naked single strand of circular RNA!
They
probably defy almost all of the properties that are attributed to life and yet
they are alive.
Theodor
Otto Diener, a Swiss American Plant Pathologist published a paper in 1989 in
the PNAS USA where he proposed them to be the most plausible candidates (even
more than the introns) as “living fossils” of a precellular RNA world.
Do
you recall what an intron is from your basic molecular biology course in high
school?
Perhaps
faintly and loosely.
Intron
comes from two words intragenic region, (intr + on), which means a region
inside the gene.
Introns
are intervening sequences of nucleotides inside genes that are edited out to
convert a precursor messenger RNA into a mature messenger RNA.
The
genetic term for this is RNA splicing.
Enough
of genetics and let us return back to our second-order arithmetic.
In
second-order arithmetic, the two divisions or the two sorts represent two
different things.
The
first sorts represent the natural numbers and second sorts are the class
variables that represent set of natural numbers.
The
individual terms include constant 0, the unary function S (or what we have been
calling the successor function all along), the binary operation + plus and .
multiplication.
Finally,
let me show you some of the basic axioms of the second-order arithmetic.
Let
us first see the three axioms that command the successor function and zero:
[1] ∀
This
says that the successor of natural number is never zero.
[2] ∀
This
says that the successor function is injective.
An
injective function is sometimes shown using an arrow with a barbed tail like
this ↣
I
will talk about the injective function in the nights to come.
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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