Cantor (Set Theory- countable sets of rational & algebraic numbers, uncountable sets of real numbers)

In his 1874 paper Cantor considers at least two different kinds of infinity. Before this orders of infinity did not exist but all infinite collections were considered 'the same size'. However  Cantor examines the set of algebraic real numbers, that is the set of all real roots of equations of the form

an xn + an-1 xn-1 + an-2 xn-2 + . . . + a1 x + a0 = 0,

where ai is an integer. Cantor proves that the algebraic real numbers are in 1-1 correspondence with the natural numbers in the following way.

For an equation of the above form define its index to be
|an| + |an-1| + |an-2| + ... + |a1| + |a0| + n.

There is only one equation of index 2, namely x = 0. There are 3 equations of index 3, namely
2x = 0, x + 1 = 0, x - 1 = 0 and x2 = 0.

These give roots 0, 1, -1. For each index there are only finitely many equations and so only finitely many roots. Putting them in 1-1 correspondence with the natural numbers is now clear  but ordering them in order of index and increasing magnitude within each index.

In the same paper Cantor shows that the real numbers cannot be put into 1-1 correspondence with the natural numbers using an argument with nested intervals which is more complex than that used today (which is in fact due to Cantor in a later paper of 1891).

Cantor now remarks that this proves a theorem due to Liouville, namely that there are infinitely many transcendental (i.e. not algebraic) numbers in each interval.

In his next paper, the one that Cantor had problems publishing in Crelle's Journal, Cantor introduces the idea of equivalence of sets and says two sets are equivalent or have the same power if they can be put in 1-1 correspondence. The word 'power' Cantor took from Steiner. He proves that the rational numbers have the smallest infinite power and also shows that Rn has the same power as R. He shows further that countably many copies of R still has the same power as R. At this stage Cantor does not use the word countable, but he was
to introduce the word in a paper of 1883.

Cantor published a six part treatise on set theory from the years 1879 to 1884. This work appears in Mathematische Annalen and it was a brave move by the editor to publish the work despite a growing opposition to Cantor's ideas. The leading figure in the opposition was Kronecker who was an extremely influential figure in the world of mathematics.

Kronecker's criticism was built on the fact that he believed only in constructive mathematics. He only accepted mathematical objects that could be constructed finitely from the intuitively given set of natural numbers. When Lindemann proved that is transcendental in 1882 Kronecker said

Of what use is your beautiful investigation of . Why study such problems when irrational numbers do not exist.

Certainly Cantor's array of different infinities were impossible under this way of thinking.