I think, the worst thing that can happen to mathematics is to impose some dogma onto it. And the worst kind of dogmas are those concerning foundations. It is in the essence of mathematics that an object has its existence justified as soon as it is clear that this object is well-defined. And that´s really all about it. There is no usefulness or appropriateness or 'better' when it comes down to laying the foundations. For axioms are not subordinated to their implications. As for the so called 'intuition', the latter tells me that $\mathbb{Q}$ has greater cardinality than $\mathbb{N}$...

I think, the worst thing that can happen to mathematics is to impose some dogma onto it. And the worst kind of dogmas are those concerning foundations. It is in the essence of mathematics that an object has its existence justified as soon as it is clear that this object is well-defined. And that´s really all about it. There is no usefulness or appropriateness or 'better' when it comes down to laying the foundations. For axioms are not subordinated to their implications. As for the so called 'intuition', the latter tells me that $\mathbb{Q}$ has greater cardinality than $\mathbb{N}$... In other words I don´t see a single reason why we shouldn´t have 'multiple' mathematics, each based on its foundations, each equally justified!