Today in VERITAS we investigate how the theory of sets orignated. The story of the set theory is also the story of the struggles, the genius and the torment of Georg Cantor. Today’s VERITAS is inspired by a chapter from the the wonderful book “Men of Mathematics” written by the eminent Mathematician E.T Bell.
In 1874 the 29 year old Cantor published a very controversial paper that marked the beginning of the set theory. The concept of sets was known earlier but Cantor gave it mathematical rigor and made it a branch of mathematics. Cantor however was not thinking about sets.
He was trying to prove something else. This paper was on algebric numbers. Any number( natural or otherwise) which is the solution of any algebric equation (with whole number coefficiants) is said to be an algebric number. For example take the equation:
x+5 = 0
The solution to this equation is x = -5 so -5 is a algebric number.
Now isnt it obvious that algebric numbers are more than natural numbers?! To see this we notice that the set of algebric numbers includes all whole numbers since they are solutions of x-n = 0 Also algebric numbers contains the solutions of all quadratic equations with whole number coeffs + solutions of all cubic equations with whole number coeffs+ etc etc .
But to everyone’s surprise Cantor proved that the set of all algebric numbers has the same number of elements as the set of natural numbers. He did this by forming a one to one correspondence between the two infinite sets.
In another paper Cantor showed that the number of transcendental numbers is infinitely more numerous than the number of natural numbers. This did this again by doing a one to one correspondence between the two sets. THis result was a shocker!!! Mathematics had still not proved that transcendental numbers exist!
Cantor was not working on a set theory but his methods of investigations into sets of infinite numbers created the mathematical theory of sets.
A denumerable set is a set whose elements can be counted as 1,2,3 etc. So a denumerable set has less than or equal numbers as the set of natural numbers. A non-denumerable set has more numbers than a set of natural numbers. Do non-denumerable sets exist? Cantor proved that they do. He also showed that the set of points in any line segment(no matter how small) is non-denumerable.
Note that the set of natural numbers has infinite numbers. So a non-denumerable set is also has infinite elements but the number is more infinite than the set of natural numbers.
So Cantor was dealing with more infinite and less infinite and infinitely infinite in his theories. WoW!
He proved these results and several others in a set of papers from 1879 to 1884. These papers started the theory of sets.
Several mathematicians and the great Kronecker in particular rejected Cantor’s mathematical theories. Kronecker was particularly against his methods of reasoning, his treatment of dealing with infinities and his theory of transcedental numbers. “How can anyone prove that transcedental numbers are infinitely more than natural numbers when we still doubt their existance”. But Cantor’s proof did not require that transcedental numbers exist!
Kronecker was an extremely influential mathematician and made Cantor suffer a great deal by convincing other mathematicans that Cantor’s work is nonsense.
Cantor was a very sensitive man and had several mental breakdowns. He would spend some years in mental asylum , get cured and do more math and Kronecker would hate him more and Cantor would go back to the asylum.
The problem with Cantor was that Kronecker’s criticism would make him doubt his own work and he would drive himself crazy.
But by the turn of the century Cantor’s theory of sets became acceptable to mathematicians and his work came to be regarded as the work of a genius.
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