January 21, 2017 Saturday
Bedtime Story
After Zermelo, Enter Abraham Fraenkel and Thoralf Skolem
As I had mentioned earlier, towards the end of 1800s and early
1900s many mathematicians beside Hilbert were very unhappy with the state of
affairs in mathematics.
Paradoxes were popping up everywhere much to the chagrin of them
all.
Peano was amongst the first of the few to start the process of
restructuring of mathematics.
Zermelo later tried to get rid of them with his set theory.
Yet warts remained.
As Ernst Zermelo tried to cleanse the set theory of the Russell’s
paradox, a compatriot of his by the name of Abraham Fraenkel pointed out some
residual flaws in his axiomatic system.
Abraham Fraenkel was a fervent admirer of Cantor and his
path-breaking work on set theory.
So passionately impressed was Fraenkel with Cantor and his work
that he went on to write a biography on Cantor.
Fraenkel (along with Skolem) wrote a letter to Zermelo in 1922
stating that his axioms could not prove the existence of certain sets and
cardinal numbers.
Thoralf Skolem was a Norwegian mathematician who made significant
contributions to both set theory and mathematical logic.
In other words, Zermelo’s axioms were not able to prove the
existence of the set {Z0, Z1, Z2…} where Z0
is the set of natural numbers and Zn+1 is the power set of Zn.
The correspondence through letter writing between Fraenkel and
Zermelo started in 1921 wherein Fraenkel spoke about his new axiom schema of
replacement.
This axiom was soon discovered independently by Thoralf Skolem.
The axiom schema of replacement states that the image of any set
under any definable mapping is also a set.
Let us say if you define a class function F for a set that
projects the images of its elements, then the image set F[A] is also a
set.
Fraenkel along with Zermelo came up with new set of 9 axioms that
did not include the axiom of choice.
It is a pity that while Fraenkel and Zermelo got the credit,
Skolem was left out.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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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, I
recommend this large collection of Halloween Songs for Kids:
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