> For instance, set theory as the foundations of mathematics is a pretty solid model but it is seen as fundamentally unsatisfying for many reasons -- most famously the continuous hypothesis.

To confirm, s/continuous hypothesis/continuum hypothesis/ ?

Also, for the curious, here's a paper pushing back on the idea that the continuum hypothesis et al. even "need" to be resolved in the first place: https://arxiv.org/abs/1108.4223 (The set-theoretic multiverse, JD Hamkins, 2011). (I don't know anything about set theory myself, so I can't personally comment on what it says—but AFAICT Hamkins is respected in the area.)