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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.

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closed as not constructive by Thomas, hardmath, rschwieb, Norbert, Pedro Tamaroff Nov 14 '12 at 1:41

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Thinking about sets and infinities intuitively fails? No. That depends entirely on one’s intuition. –  Brian M. Scott Nov 14 '12 at 0:01
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Something to keep in mind is that the idea of "topology being based on the notion of (open) sets" is rather modern. Reading the literature of the period, you will see that researchers were still debating whether the real line was made out of points. –  Andres Caicedo Nov 14 '12 at 0:01
    
@BrianM.Scott So you are saying you were born with a full understanding of all the axioms of ZFC and a notion of a class, for example? –  glebovg Nov 14 '12 at 0:02
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Of course not; intuition in any area of life improves with training. There’s nothing wrong with thinking of sets as collections of objects. I doubt that you’ll find many applied mathematicians or physicists who think that the real numbers include infinitesimals. On the other hand, infinitesimals do exist in the hyperreals, and they behave very much as one would informally expect them to behave. –  Brian M. Scott Nov 14 '12 at 0:17
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I don't think people are opposed to Cantor's ideas as they are opposed to radical change. Many well known mathematicians delayed the publishing of their work for fear of controversy (Newton with Principia, Gauss with non-Euclidean geometry). This seems to suggest that there is just some inherent resistance towards paradigm shifts. –  EuYu Nov 14 '12 at 0:26