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It seems pretty strange to me that the idea of infinity and infinite sets was initially opposed by many prominent Mathematicians, even driving Cantor into depression. However, in modern days, everyone seems to accept infinity without any problem, despite the existence of many paradoxes like the Banach-Tarski paradox and the prisoner hat paradoxes.

I understand that modern Mathematics is largely built on the concept of infinity, and Mathematics has already become a "game of ideal world," and I am not here to oppose the idea. What I am confused about is that historically once upon a time the majority of Mathematicians opposed the idea (at a time where Mathematics was not yet built on infinity), even driving Cantor himself into craziness. It seems pretty strange to me why and how the world has changed to this current state, where anyone opposing the idea is labeled a "crank".

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    $\begingroup$ Who exactly opposed the idea of infinity at the time of Cantor? Afik, people opposed to his set theory on different grounds, not on his usages of infinite sets? $\endgroup$ Aug 9, 2020 at 19:25
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    $\begingroup$ Also, "modern Mathematics is largely built on the concept of infinity" is not quite right. Most of modern math uses set theory and infinite sets (not "infinity" which is ill-defined) are prominent features of ST that very few working mathematicians object to. $\endgroup$ Aug 9, 2020 at 19:33
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    $\begingroup$ The idea of set is much younger than the idea of infinity. $\endgroup$ Aug 10, 2020 at 2:54
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    $\begingroup$ Cantor's theory incorporated infinite sets and worked much better in analysis and algebra than its predecessors. It also answered traditional objections, like the "annihilation of the finite" in arithmetic. This is no more pure opinion or less objective and "firm" than Copernican system explaining astronomical data better than Ptolemaic, or calculus serving mechanics better than traditional geometry, or predicate logic being more expressive than Aristotelian. Measuring the Size of Infinite Collections by Mancosu is a good historical sketch. $\endgroup$
    – Conifold
    Aug 10, 2020 at 5:39
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    $\begingroup$ As I said, people objected to Cantor’s set theory, but it was not due to his use of infinite sets. The objection was to his level of abstraction. Infinite objects, not yet called sets, were widely used since introduction of Calculus. I think, your real question is about acceptance of Cantor’s set theory, not the idea of infinity. $\endgroup$ Aug 10, 2020 at 16:08

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Quoting numerous sources,

The German physicist Max Planck said that science advances one funeral at a time. Or more precisely: “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”

That's not 100% applicable here, since mathematics simply accepts anything which results from a valid proof. Remember that math does not necessarily hew to the physical world, so the existence of "infinities" of different measure is perfectly acceptable, just as is nonEuclidean geometry, the Cantor set, and the Discrete Metric, to name some famous examples.

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  • $\begingroup$ "math does not necessarily hew to the physical world," this viewpoint was not common once upon a time. I know it is the way it is now, but historically it seemed more probable to become a different universe where Mathematics is built on a more real-world based framework. $\endgroup$
    – cr001
    Aug 10, 2020 at 14:44
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It's not that surprising when all things are considered. Most people do not like their cherished ideas over-turned. Take, for instance, calculus. Newton expected opposition to his new ideas, which is why he couched his discoveries not in calculus, but in the mathematical language of the day: Euclidean geometry. Given his mathematical sophistication, no doubt he actually enjoyed coming up with geometric proofs of what he discovered by use of the calculus.

Set theory was still a new idea when Cantor came up with his new theory of the infinite. So for most mathematicians, this was newness piled upon newness - so probably one newness too many. Still, most mathematicians came round to Cantor's point of view, since his reasoning was sound. The main problem with it is that it is a theory with very few real applications.

In other words, it's widely accepted but also widely useless.

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