November 27, 2016 Sunday
Bedtime Story
Cantor's Uncountability Theorem
Cantor's Uncountability Theorem
Cantor makes the following remark after stating his second
theorem:
“The reason why collections of real numbers forming a so-called
continuum (such as all real numbers which are ≥0 and ≤1) cannot correspond
one-to-one with the collection (v) [the collection of positive integers];
Thus I have found a clear difference between a so-called continuum
and a collection like the totality of real algebraic numbers.”
This is Cantor’s uncountability theorem.
In the paper, Cantor did not actually prove it but merely stated
it.
It should be emphasized that a benign statement such as:
“The set of real algebraic numbers can be written as an infinite
sequence in which each number appears only once.”
needed an immense effort from many mathematicians to be proved.
Cantor himself could not prove it.
The proof would come from another German mathematician and perhaps
the only mathematician in the early days to both appreciate and admire Cantor’s
work.
It was Richard Dedekind who met Cantor accidentally on a holiday
at the beautiful town of Interlaken situated in the Bern canton of Switzerland
in the year 1872.
Dedekind was the first mathematician to give a precise definition of
an infinite set and real numbers.
With what goes now by the name of Dedekind cut, he gave a whole
new perspective of looking and constructing real numbers.
It is a beautiful idea and I would like to share it with you.
He considered the famous number line that contains all the
rational and irrational numbers.
If you cut the line anywhere with an irrational number, you end up
creating 2 non-empty sets A and B of rational numbers such that all the elements
of A are less than all the elements of B.
That seems very apparently obvious.
What is less obvious is this next statement:
Moreover, the set A will contain no greatest element of rational
number.
This statement baffles me till this date.
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:
On the left hand side of the root 2, there is no greatest element. For there will always be yet another irrational number left of root 2.
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