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Thanks a lot for the very comprehensive answer and links. I guess the answer to my question would be "The cardinality-challenged mathematician would get along just fine since a very large part of 'regular math' can be done with weaker versions of set theory without crazy multiplicities of uncountable cardinals" Large cardinal theory sounds very intriguing. Are there any good books at the advanced undergrad/beginning grad level I could learn it from ?
Terry: "To me, it's not so much that the multiplicity of uncountable cardinals comes up directly all that much in mathematics, but rather that they are inevitable byproducts of the very useful set theory axioms" Thanks ! That's very helpful.
Terry, I was coming at this from a different angle. For instance, suppose we did not have the Power Set Axiom. One could still prove that the reals had a higher cardinality than the naturals as in the link below for example. boolesrings.org/scoskey/my-favorite-proof-that-r-is-uncountable Would there be any theorems outside of set theory - hence, not relying on the Power Set Axiom, perhaps - that would force us to conclude that some mathematical object had a cardinality higher than R ?
