December 03, 2016 Saturday
Bedtime Story
Cantor contemplates comparing the Infinity of the Integers with that of the Real Numbers
Cantor contemplates comparing the Infinity of the Integers with that of the Real Numbers
He further went on to prove the equivalence of the set of the real
numbers
and the n-dimensional
Euclidean space Rn.
Kronecker was deeply upset with these heretical results that
Cantor kept on producing and tried his best to make Cantor withdraw his papers
from the Crelle’s Journal.
Dedekind on the other supported Cantor and so did Karl Weierstrass
(both Kronecker and Weierstrass were editors of this journal in 1880s and
Dedekind was not).
Cantor had submitted this paper of his but then onwards, not a
single paper did he sent to Crelle’s Journal, very embittered and discouraged
by the editor.
Cantor then on went to work on a novel concept that goes by the
strange name of continuum hypothesis.
By now, you must be well aware of the essence of the word
continuum.
Continuum in the set theory of mathematics refers to real numbers
whose cardinality is denoted by the lower case script letter
.
Unlike integers, they are very literally continuous on the number
line and hence the name.
The cardinality of a continuum is the size of the set of real
numbers which is infinite.
That any kid can tell.
Cantor put forth his continuum hypothesis in 1878, not far removed
from his landmark first paper of 1874.
By then, Cantor’s uncountability theorem had been proven; that the
cardinality of the set of integers (aleph naught or
0) is less than the cardinality of the set of real numbers.
But by how much?
Imagine the kind of mind this man had!
To ponder over such abstract ideas which average mortals like us
would deem inefficacious and which geniuses like Galileo simply failed to
grapple and abandoned them.
To answer it, Cantor came up with this remarkable continuum
hypothesis.
This hypothesis says that the next greatest infinity after the
infinity of integers is the infinity of the real numbers.
We will discuss this extraordinary hypothesis in the nights to
come.
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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