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When reading about some open problems, a lot of them have quotes by renowned mathematicians that “[the conjecture] cannot be solved using the current technology” or something along these lines. What do they mean by that? Are they talking about the axioms? Or are they generally speaking in terms of intelligence and mathematical abilities?

These ones are just at the top of my mind:

https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/

https://m.youtube.com/watch?v=MXJ-zpJeY3E (skip to the end where the talk about the Riemann Hypothesis)

But I’ve seen many, many more examples