This story fits into the larger story of mathematical Platonism, in which mathematical truths exist eternally and universally, independent of humans, and mathematicians are brave explorers and discoverers.
The counter-story, of humanistic mathematics, instead focuses on the context of mathematical results: the communities of practitioners -- with their established methods and ceremonies, their modes of thought, and their tacit knowledge -- and the larger societal context in which these mathematicians work. In this story, mathematics is a part of culture, like music or politics, and it is created by humans.
Both stories capture valid aspects of mathematics, and it is interesting that the story of Cantor and this particular result can be fit into either. The traditional telling, as mentioned above, threads Cantor's discovery into the Platonic story of mathematics. But a close reading of Cantor's correspondence with Dedekind (see, for example, Fernando Gouvea's 2011 article Was Cantor Surprised?), allows for a humanistic interpretation, as we now explain.
Instead, to get around this Cantor responded, after three days, with a completely different proof, involving several elaborate and technical steps. After four days Cantor was impatient, and wrote again, saying about the tricky steps of the proof, "I can have no peace of mind until I obtain from you, honored friend, a decision about their correctness. So long as you have not agreed with me, I can only say: Je le vois, mais je ne le crois pas."
"Seen in its context, the issue is clearly not that Cantor was finding it hard to believe his result. He was confident enough about that to think he had rocked the foundations of the geometry of manifolds.
"Rather, he felt a need for confirmation that his proof was correct. It was his argument that he saw but had trouble believing."
— Fernando Gouvea