August
07, 2017 Monday
Bedtime
Story
Third Axiom of Second-Order Arithmetic
Third Axiom of Second-Order Arithmetic
Last
night I had let you with the second axiom of the second-order arithmetic.
It
says that the successor function is injective.
The
injective function preserves distinctness.
So
what this is saying that if there exists two natural numbers m and n, then
their successors will be two natural numbers that will be distinct.
[3] ∀
This
axiom says that every natural number is either a zero or a successor.
I
will not go too much into the second-order arithmetic as I merely wanted to get
a feel of it.
You
may wonder what big difference there is between first-order and second-order
arithmetic or logic.
That
would be a very valid question.
One
of the key distinctions between the two is that the first-order arithmetic uses
variables that have range over individuals.
Individuals
refer to the elements of the domain of the discourse.
Second-order
arithmetic besides having these variables also has in addition variables that
range over sets of individuals.
Real
numbers can be represented as infinite sets of natural numbers.
Since
second-order arithmetic allows quantification over such sets, the real numbers
can be formalized in second-order arithmetic which is not possible in the
first-order counterpart.
Some
mathematicians see second-order arithmetic as weak version of set theory (much
weaker than Zermelo-Fraenkel set theory) where every element is either a
natural number or a set of natural numbers.
Almost
all the results of classical mathematics can be proved via second-order
arithmetic.
Now
that we are a little bit more enlightened about the second-order arithmetic,
perhaps the statement below that is arrived from Tarski’s theorem will make
more sense.
A
truth predicate for the first-order arithmetic can be defined in second-order
arithmetic.
It
is evident from this discussion then that there has to exist some sort of
arithmetical hierarchy.
Two
mathematicians who are associated with this concept of arithmetical hierarchy
(and hence recursion theory) are Stephen Cole Kleene (American) and Andrzej
Mostowski (Polish).
I
will not delve either into the topic of arithmetical hierarchy nor in the story
of these two brilliant mathematicians who are better known for laying the
foundations of the recursion theory rather than of arithmetical hierarchy
though both these are related.
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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