Cantor's fortune

Question

Wiki says that his transfinite numbers met opposition:

Henri Poincaré referred to his ideas as a "grave disease" infecting the discipline of mathematics, and Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong"

and

Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism

Can you briefly explain when the commotion started, if it regarded also his work on number or set theory , show what was the most grounded or shared criticism and when the opposition ended and who turned the tide? Was it Hilbert ? Could you please specify if any doubt is still surviving on any aspect of his theories?

EDIT

This is definitely not a duplicate of the linked question nor the links in the comments provide any help:

That question tackles a rather marginal problem: whether Poincaré said it or not? I asked what other ideas met opposition and when / by the help of whom criticism was overcome and, indeed, if it has definitely or if even today there are some detractors?

Answer 1

This question is too broad, and it is impossible to give a short answer.

In the beginning of 20th century, Cantor's discoveries led to what is called a "crisis of foundations" in mathematics. You are right. Hilbert was one of the principal defenders of Cantor's theory, but paradoxes were soon discovered in set theory and the heated discussion continued.

There were several programs on how to get rid of these paradoxes, the most famous of them being "Hilbert's program". Two other alternatives were Intuitionism and Type Theory. Later, Gödel showed that Hilbert program would not work. For the full story, I recommend an excellent book:

Fraenkel, Abraham A.; Bar-Hillel, Yehoshua Foundations of set theory. Studies in Logic and the Foundations of Mathematics North-Holland Publishing Co., Amsterdam 1958.

One can say that the discussion on the foundations of set theory still continues, but most mathematicians are satisfied with ZFC axioms.