December 06, 2016 Tuesday
Bedtime Story
Resolving the Antimonies or the Paradoxes
These men, particularly Zermelo also came across what now goes as
Russell’s paradox in the set theory that I had dealt with some nights ago.
Zermelo decided the tackle this problem in set theory by
formalizing it as he considered Cantor’s approach to be naïve (naïve being used
in a technical sense and is known as Cantor’s naïve set theory or NST).
In a paper that he published in 1908, titled “A New Proof of the
Possibility of Well-Ordering” he listed a set of axioms that would take care of
the antinomy that kept arising in Cantor’s naïve set theory.
Antinomy is a terminology often used in logic that reflects a real
or apparent mutual disagreement or contradiction of two laws.
Antinomy is very similar to paradox and the difference is really
hard to define.
Antinomy may differ from the paradox in that antimony involves 2
laws that apparently contradict each other.
In the paper Zermelo proposed 7 axioms which I shall list out with
very brief explanations.
The axioms though stated in simple English (and at times seem to
be childishly simple) are very technical and not easy to grasp unless you
really understand the set theory and the antimonies or paradoxes that lie
within it.
The Seven Axioms of Zermelo set theory are:
1. Axiom of extensionality:
Consider two sets A and B. A is equal to B if and only if both
have set X as their elements.
In short, two sets are equal if and only if they have precisely
the same members.
This axiom involves predicate logic which I shall go into a bit
later.
2. Axiom of elementary sets:
Consider two sets A and B.
There is a set C such that given any set D, D is a member of set C
if and only if D is equal to A or D is equal to B.
In other words, given two sets, there is a set whose members are
exactly the two given sets.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night and my fellow cousin ape.
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, may I
suggest this large collection of Kids Songs:
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