January 15, 2017 Sunday
Bedtime Story
Peano axioms
Second-order logic or the propositional logic is an extension of
the first-order logic that uses quantifiers both for variables and
relationships.
Second-order logic has variables and
quantifiers not only for individuals but also for subsets of the universe.
It is not easy for me explain second-order logic further than this
though the last axiom is an example of that.
Now let me list out the nine axioms of Peano:
1. 0 is a natural number.
2. For every natural number x, x = x. (Reflexive relationship)
3. For all natural numbers x and y, if x = y, then y = x.
(Symmetric relationship)
4. For all natural numbers x, y and z, if x = y and y = z,
then x = z. (Transitive relationship)
5. For all a and b, if b is a natural number and a = b, then a is
also a natural number.
6. For every natural number n, S(n) is a natural number.
7. For all natural numbers m and n, m = n, if and only if,
S(m) = S(n). (Injective function)
8. For every natural number n, S(n) = 0 is false.
(In other words, there is no natural number whose successor is 0)
9. If K is a set such that:
0 is in K and
For every natural number n, n being in K implies that S(n) is in
K,
Then K contains every natural number.
This last axiom is known as the axiom of induction and is an
example of second-order logic.
This axiom of induction is most interesting one and needs some
more investigation.
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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