November 25, 2016 Friday
Bedtime Story
Basic Notations Devised by Cantor
Basic Notations Devised by Cantor
Consider the infinite set of natural numbers:
N = {1, 2, 3, 4, 5…} has the least element as 1.
So it is a well ordered set with a total order (because 1, 2, 3
and so on obey the conditions of antisymmetry, transitivity and totality) and
has the least element 1.
Every non-zero natural number has a unique number that precedes it
(For example, 99 will always and only precede 100).
Another ordering in it would be that all the even numbers are less
than the odd numbers (2 < 3, 4<5, 6<7 and so on).
Now consider the infinite set of integers:
Z = {…-3, -2, -1, 0, 1, 2, 3, 4…}
Now Z has all the conditions fulfilling total order but since it
has no least element, it is not a well ordered set.
The ordinal number deals only with well-ordered sets.
Another idea that Cantor developed was that of the order type of
the set.
The order type gives the length of any well-ordered infinite set,
the smallest infinite ordinal being represented by the Greek small letter omega
ω.
The natural numbers as you will intuitively understand is the
smallest totally ordered set with no upper limit or no upper bound.
The first infinite ordinal is ω and that comes after all the
natural numbers.
With this, I have tried to define the basic technical terms that
were coined by Cantor to develop his astounding ideas.
They are:
1. One-to-one correspondence or bijection
2. Transfinite numbers (numbers that are more than finite but less
than uncountable infinite)
3. Cardinal numbers or cardinality of infinite sets which is
denoted by the symbol aleph 
and which can be 
0, 1, 2…
and which can be 0, 1, 2…
The natural numbers are infinite but countable and have been
assigned the lowest cardinality 0.
0.
The cardinality of the real numbers 
also known as the cardinality of the continuum
is denoted by the letter 
which is the lower case letter of Fraktur
script.
also known as the cardinality of the continuum
is denoted by the letter which is the lower case letter of Fraktur
script.
4. Ordinal numbers represented by the symbol ω and the idea of total-order
and well-ordered 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:
The first is the Hebrew letter aleph and the second is the lower case of fraktur script "c" which is used to show the size of the set of Real numbers
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