I know we have accepted Cantor's ideas a long time ago and many mathematicians use sets and infinities without ever realizing that thinking about sets and infinities intuitively fails, because there are many paradoxes associated with naive set theory. However, why did mathematicians such as Kronecker regarded Cantor's ideas as absurdities (and as I remember accused Cantor of impiety and the corruption of youth)?
Also, I believe Poincare did not like Cantor's ideas, yet he did a lot of research in topology, which is based on the notion of an open set. Can any one explain why so many people apposed Cantor's ideas? Why is the traditional view of infinite so appealing although Cantor's proofs are valid?
I posted this question back in 2015, and since then I have changed my views on the validity of Cantor's arguments. I no longer believe they are valid arguments. Many people nowadays just have "itching ears" so to speak and they want to turn away their ears from the truth, and turn to fables. Cantor's theorems are fables; they are hearsays because many mathematicians just repeat them without taking time to consider what they are teaching their students.
I have studied calculus and analysis and I know one can study it and prove theorems without even mentioning Cantor's ideas. You can completely avoid the modern definition of an infinite set. The modern definition of an infinite set is so "strange" that it creates paradoxes, and people have to constantly add more axioms to the Zermelo-Fraenkel theory to fit a square peg into a round hole. For example, instead of saying "x belongs to the set of real numbers" one can just say that "x is a number such that the square of x is greater or equal to zero" to avoid defining the set of real numbers. This is how people have thought of real numbers for centuries. Moreover, in calculus and analysis, we never actually use infinity: look up every definition in calculus and you will only find a symbol for infinity. For example, look up the definition of a limit "at infinity" and you will find that there is no mention of infinite sets or cardinals.