November 26, 2016 Saturday
Bedtime Story
The 1874 Paper of Georg Cantor Contained the First Theorems of Transfinite Set Theory
The 1874 Paper of Georg Cantor Contained the First Theorems of Transfinite Set Theory
Remember, Cantor was more interested in numerical sets that were
infinite and until him, no one even knew how to tackle them.
Anybody who had handled them earlier had landed up in paradoxes
and left things at that.
Mathematicians consider the beginning of modern set theory with
the publication of Cantor’s seminal paper in 1874 titled “On a Property of the
Collection of All Algebraic Real Numbers”.
This paper is also known as Georg Cantor’s first set theory
article and is rather short for its greatness.
It spans a mere 4 full pages and 1/3rd of another.
At the very onset, Cantor in his paper states his first theorem.
The set of real algebraic numbers can be written as an infinite
sequence in which each number appears only once.
It does not seem to be very extraordinary theorem at first glance,
not even at the second well-pondered glance.
What Cantor was implying in his first theorem is that the entire
set of real algebraic numbers can have one-to-one correspondence with the set
of positive integers.
Following this, Cantor goes on to state the second theorem which
pertains to numbers in a closed interval of a and b where a and b are certain
integers with a ≠ b:
Given any sequence of real numbers x1, x2, x3…and any interval [a,
b], there is a number in the interval [a, b] that is not contained in the given
sequence. Hence, there are infinitely many such numbers.
That is not very difficult to imagine.
Let us consider say a set of positive integers {4, 5} only.
The let us consider a set of real numbers in the interval [4, 5].
The real numbers between 4 and 5 can be {9/2, 19/4, 23/5, 24/5…)
and many more.
It is evident that there does not exist a one-to-one
correspondence between the set of positive integers {4, 5} and the set of real
numbers lying in the interval [4, 5].
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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