To be completely honest, this looks like what I like to refer to as “symbol vomit”. And also, the square root of 0 is just 0, that is the definition you will find almost everywhere, so there is no need for this weird symbol salad. As for the author, I couldn’t find him apart from like two of his books, do you have any more infos on him? Because this looks very non-mathematical apart from the symols.
This has nothing to do with CH. In the middle cone there are counting numbers (ordinals to be precise), with the numbers omega (the smallest counting number which is bigger than any natural number, like an infinity-th counting number), omega+1 (the counting number after omega, a bit like infinity+1) and omega_1 (the first counting number which has no 1-1 correspondence with omega) marked explicitly. The alpha may be replaced by any infinite counting number.
The R(alpha) at the side are just some examples, how far “the universe” (something properly defined in the book) has to go up to to do reasonable set theory. Those R(alpha) are just the sets (think of them like nice enough collections of objects) which can be constructed in a finite amount of steps from the set containing nothing.
Also, as for the second to last paragraph in your comment, it is known that CH is independent of ZFC, the axioms most commonly used for set theory.
This depends on what properties you want your number system to satisfy. Usually you want for any three numbers a,b,c to satisfy
Associativity of addition: a+(b+c)=(a+b)+c This is quite useful, so we don’t want to give this up
Commutativity of addition: a+b=b+a Also useful but you could get around that if you really want to, but for our purposes let’s keep it
An additive identity (or zero): 0+a=a=a+0 You want a zero, so this is needed
Additive inverses: There exists x such that a+x=0 (here x=-a); you also want this
Associativity of multiplication: a*(bc)=(ab)*c Same as above, you want this property
Commutativity of multiplication: Useful but not necessary
A multiplicative identity (or one): 1a=a=a1 Usually with 1=/=0, also useful
Multplicative inverses for nonzero elements: Not that necessary, there are useful number systems without this (like the integers …,-1,0,1,…)
Distributivity: a(b+c)=ab+ac, (a+b)c=ac+bc You ant this, as it links addition and multiplication and this is quite desirable.
If you assume 4. and 9., you get 0a = (0+0)a=0a+0a, hence 0=0a. This means that you would have to give up distributivity wihin your number system, however distributivity is what links addition and multiplication together, hence your question would just be “what if we have two binary operations that don’t really interact with each other?” and the answer is: Maybe there are useful cases?
Edit: I forgot about losing property 4, in which case some examples are found in the following math stackexchange post