I’m not even close to being an expert in mathematics, but I have some ideas regarding the continuum hypothesis and the generalized continuum hypothesis.

CH

Regarding the continuum hypothesis, I think one possible way to have a cardinality between ℤ and ℝ might be to define something like the real numbers but where certain elements of the sequence composing the number or patterns of elements are equivalent to certain other elements or patterns of elements.

For example, you specify your numbers in base 10 and define that all 8’s are equivalent to 9’s, so 0.7652861066492217 would be equal to 0.7652961066492217, 0.7652861066482217, and 0.7652961066482217. (Not sure if it would become an issue that 1=0.999…=0.888…, 2=1.999…=1.888…, etc., so you might not be able to have any whole numbers, but that’s just an artifact of this particular example.:P Also, making one numeral equivalent to another is probably the same as working in base 9 instead of base 10, which shouldn’t impinge on its cardinality. So maybe that’s a bad example, or maybe this disproves my whole tack.)

Another example would be to use binary and define that all sequences of {0, 1, 0} are equivalent to {0, 1, 1}. So 0.3125, or {{}, {0, 1, 0, 1}} would be equivalent to 0.4375, or {{}, {0, 1, 1, 1}}. (In this notation I’m defining real numbers as sequences of two sequences, where the first sequence is the integer part and the second sequence is the fractional part. Also, a better example wouldn’t allow for numbers/sequences where you have to choose between multiple paths of substitution. Not sure that’s even possible in base 2.)

This would create a set that’s smaller than the real numbers but larger than the integers. I have a hunch that you’d need to define your equivalencies in a way where it they happen an infinite number of times in the set of all real numbers.

GCH

Regarding the generalized continuum hypothesis, my idea is that an integer is a sequence of digits, while a real number is two sequences of digits, and in between any two consecutive integers is an unlimited number of fractional parts of the numbers, where the fractional part is the second sequence. So, naturally, to get to the next cardinality, you’d just have to add a third sequence. Between any two consecutive real numbers would be an unlimited number of possible values for the third sequence. So basically the set of this type of number is an infinite set of infinite sets of infinite sets, and it’s all ordered. For ℵ₃ you’d need numbers comprising four sequences, and so on and so on.