I should have been more specific; I understand why it was a mathematical breakthrough. What I don't understand is why it would have triggered some kind of psychological horror or philosophical crisis. It was a new way of understanding numbers, but it didn't reveal numbers to be acting any differently than we had always assumed.

If anything, it seems like it would have been comforting to finally have mathematical constructions of the real numbers. It had been disturbing that our previous attempts, the rational and algebraic numbers, were known to be insufficient. The construction of the reals finally succeeded where previous attempts had failed.

Because painting those who objected to these definitions of mathematical infinity as "horrified" and "disturbed" was a form of character assassination, which was not uncommon at the time. The high moderns didn't play.

History only seems obvious in retrospect

I would invite you to be more open to the idea that people don’t live in a world where they operate inside a theoretical framework with localized test actions

major breakthroughs tend to cause existential crises because most people don’t have full scope of their work in order to understand where it is broken