Even if you're willing to accept the idea of cardinalities of infinity (and I think you ought to), I find that the more broad acceptance of 1. The axiom of choice (vs the axiom of determinancy, it's opposite) and 2. The law of excluded middle to be highly suspicious.
If you reject the axiom of choice, you're just in alternative but still correct math.
If you reject the law of excluded middle, you've ended up with intuitionistic or "constructive" logic, where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Notably, in intuitonistic/constructive logic, infinity is rejected until it can be "constructed", which also means that Cantors diagonalisation argument is not so naively accepted. While that diagonalisation itself was constructive, Other related and stronger theories from Cantor are not.