I see a close relation between some of the basic questions of physics and the various definitions of continuum hypothesis.
First I mention the problem called quantum gravity. Serious difficulty is that some quantities become infinite in general relativity and quantum physics as well. An even greater challenge is creation of unified theory. I do not want to analyze the more successful and less fruitful attempts of unification. Instead, I'll relate these inefficient efforts with a mathematical question. Seemingly1 the most diverse properties of general relativity and quantum physics are their dealing with space and time, I think, and these attitudes are linked to the fact that one of them describes the enormous universe, and the other explains the micro-world.
In mathematics, there are two concepts - extensive and intensive infinity - which over large constructions have been establishing in mathematics, and they are seemingly independent from each other. Cantor’s transfinite numbers are not really harmonized with old-established and new methods2 of infinitesimal calculation.
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1 I designate this as seeming difference because space and time appear as classic background in the quantum physics, but this view is misleading. Usage of complex probability amplitude implies non-classic space-time geometry in quantum physics. This, however, is not yet known because two-element numbers is not well known as models of space-time.
2 Abraham Robinson's non-standard analysis, with acceptance - but not widespread - has become extension of infinite quantities towards intensive infinity. This is the new concept of infinitesimals.
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