Yeah I’ve always disagreed with Georg Cantor. Though his thought experiments are infinitely fun. Here’s one reason among infinite others. One argument for more than one set of infinities is the following: A: if between the natural numbers ‘1 & 2, there are infinite decimal points’ and B: between the natural #’s 2 & 3 there are infinite decimal points, then it follows C: that between the #’s 1 & 3 there should be twice as many decimal points as there are between either 1-2 or 2-3, making one set of infinite decimals between 1-3 twice as big as the infinite set between 1-2 or 2-3. This seems like a paradox, because how can be infinity be bigger than another? In fact, I propose that it’s just a logical error and grammatical confusion. For one thing, infinity is never a set, or else it wouldn’t be infinite. Infinity (I hesitate to call it ‘the infinity’ because it would again leads to the same confusion) between 1-3 is not any bigger than that between 1-2. They are both portals that once delved into, lead to (in)finite and endless sub-divisions. What’s different is the size of the portals (doors) that open to this same unitary abyss of infinity. The size of the 1-3 portal is twice as big as the 1-2 portal (imagine them as elevator doors.) But once you open the elevator door, the drop is still the same infinite shaft, regardless of of the size of the door.
A. Darius Kamali