It seems like a rant. Here are my thoughts on what they say.
Your introductory Real Analysis textbook does not make complete sense and suddenly mathematics is so counterintuitive.
Most students have felt this way, I think. My introductory Real Analysis textbook even said in the foreword that calculus made what you thought was difficult easy, while real analysis will make what you thought was easy difficult. But I have a feeling this is not what he's talking about.
Modern mathematics is based on Cantor’s infinite set theory and the concept of an actual infinity
Not Cantor's theory per se, although he did pioneering work in the understanding of the concept of infinity.
It is full of counterintuitive results and seeming contradictions
This I can agree with. That doesn't make it wrong. Physics is full of counterintuitive results and seeming contradictions, and many of those are verified by experiments, so you know they must be real. We can't experiment in the same way with math, but we can still accept oddities the same way.
we don’t know the exact numerical value of a single irrational number
If we're going down this path, what does it even mean to know the exact numerical value of something? In what way do we know the exact numerical value of $\frac13$, but not $\sqrt3$?
Irrational numbers must be denoted with symbols
So does any other number. Have you forgotten that "$2$" is a symbol as well?
leading to the absurdity that almost all members of the set of real numbers cannot be written as numbers
This seems hopelessly bound to the decimal system (and its cousins) as the numbers. It's the most common way to describe numbers, sure, but that doesn't mean it's the alpha and omega of numbers.
Two infinities with Cantor’s Diagonalization Argument, and infinite infinities with Cantor’s theorem, how can all this be true?
I don't know about true. It follows from axioms that most of us accept using logic rules that most of us accept. If you don't like Cantor's result, you must at some level disagree with some of those axioms or logic rules. Which is fine.
The Law of Excluded Middle and proof by contradiction has been misapplied to conclude that the collection of natural numbers is an infinite set.
And the Law of Excluded Middle (and the "proof by contradiction" method derived from it) is probably the most common logic rule to disagree with. Denying that rule gives you what is called intuitionistic logic. And much math has been done in that framework. But one isn't more true than the other. They are just different.
These two wrong assumptions lead to a series of cascaded incorrect results
As I said, it's not wrong. But it is true that proof by contradiction is a fruitful method, and it is "contagious" in that results proven using contradiction is then again used in new proofs, until much of the math we use every day has some contradiction somewhere backing it up.
The empty set cannot be a set
In standard ZF set theory, the existence of the empty set as a set is proven using only two axioms (and no contradiction). If you disagree with either of those there isn't much math you can do, honestly.
the existence of the Power set does not follow from the existence of a set
This is true. The existence of the Power set is its own axiom. You are free to disagree with it. Indeed, the Power set axiom is the only axiom that allows larger infinities to be constructed from smaller ones, so without Power sets, you can't prove there is more than one infinity. Score for the authors. However, the Power set axiom is also necessary for a lot or reasoning involving subsets, so until they find an alternative, I'll stick to what I know.
The axiom of infinity in mathematics is a flawed assumption
It seems they have fundamentally misunderstood what axioms are. Axioms are usually introduced as "Small results so obvious they don't need to be proven", or something like that. But that's not it at all. They are assumptions. Nothing more, nothing less. By assuming an axiom, you are not saying that you think it is true. You are merely saying you want to explore what happens if it is true. If the authors disagree with the axiom of infinity, then they are free to do that.
An object can only be potentially infinite as per Aristotle.
Because math hasn't moved forwards in 2 500 years, everything Aristotle says must be true. I'm not saying Aristotle is wrong (I don't know enough about his works for that), but accepting him as The Authority on these matters is not something I would consider a good idea.
I'm not going to comment on the physics things he says.
All in all, these are not new thoughts. But they seem radicalised in this book. Instead of being a decent book offering an alternative view on the field of mathematics, they seemingly present it as the only valid view, and anyone who doesn't accept it is wrong and stupid.