The problem is, if you are interested in the formalisation of mathematics, that there is an awful lot of stuff that patently works and is incredibly basic and useful that we simply can't prove without assuming the existence of an inductive set.

That said, without inductive sets, mathematics is far, far better behaved (and complete, which is nice). You just can't do much with it.

I didn't really get into the many infinities here. That's something more aligned to a different discussion which centers around a separate construction of numbers, called cardinal numbers in which, for the purpose of simplicity, obviously, omega is referred to as aleph zero.