Let’s start with what labels I attach to what semantics (the definitions of words I work with):
Infinity = that which has no beginning and no end both internally and externally (aka actual infinity)
Semi-infinite = that which has a beginning but no end (aka potential infinity…what mathematicians appear to be calling infinite, despite it not being infinite)
Call the set of all books x. There exists an infinite number of books. Thus x is infinite but it is not Infinity. A finite library can only finitely contain x. A semi-infinite library can only semi-infinitely contain x. That which is actually Infinite (call this Existence/Infinity) fully/completely/infinitely contains x.
Inifnity contains an infinite number of books. If only Existence/Infinity is Infinite (which It is because there can only be one actually Infinite or Omnipresent being), then only Existence fully contains x. This is a logical/semantical/definitional requirement. A semi-infinite library cannot contain an infinite number of books because by definition, it is semi-infinite (semi-omnipresent at best). Thus, a semi-infinite library cannot fully contain x. Keeping in mind that everything that exists is a member of Existence, consider the following:
x = all books in Existence (the number of which is infinite)
Finite library y contains 1 million books. Thus, y contains a part of x. That part is finite. x is infinite. y does not contain x. It contains a finite part (call this f) of x. It contains fx. There can be an infinite number of fxs. By this I mean there can be an infinite number of finite libraries. All of which, are of course, a part of Existence (they may not be as real as you and me, but they are a part of Existence. Dreams, items of thought, and hypothetical possibilities, don’t go in and of Existence because that would be paradoxical. Some attain reality, some are reflected on, some are experienced, and so on. They do not go out of Existence, nor do they come into Existence)
Semi-infinite library z contains a semi-infinite number of books. Thus, z contains a part of x. That part is semi-infinite. x is infinite. z does not contain x. It contains
a semi-infinite part (call this si) of x. Call this si of x, six. There can be an infinite number of sixs. One library can contain an endless number of red books, but it cannot contain all red books (let alone all books). This will become clearer.
Further understanding the semi-infinite:
P and Q are two identical semi-infinite libraries. We take one book away from P:
P does not have book ‘10’ despite containing a semi-infinite number of other books.
Q contains all the books in P as well as book ‘10’. Before taking out book ’10’, P and Q were identical in the number of books they contained. After taking out book ’10’, it can no longer be said that P and Q are identical in the number of books they contain. I will prove this:
Nothing can be taken in and out of Existence/Infinity. But things can be taken in and out of semi-infinite libraries. Neither libraries contain all books because neither libraries are Infinity/Existence/Omnipresent. Despite this, the libraries are connected to the Infinite. Thus, they can have access to an infinite number of books (like we can have access to an infinite number of imaginative thoughts). Libraries can have access to x, but they cannot contain x. Keep in mind, all libraries (finite or semi-infinite) are members of Existence (the set of all sets).
A and B are identical semi-infinite libraries. Where all the books in A and B are red and blue, and then all the blue books are taken out of A, then A and B are 50% identical and B is twice the size of A. Both libraries still contain a semi-infinite number of books. The semi-infinite number of books in B is more potent/concentrated, or with greater depth and breadth (just like the finite size of a 65 inch flat screen tv is greater than that of a 32 inch flat screen tv. Finite things have dimensions, so do semi-infinite things) than A. Thus, B is greater in semi-infiniteness than A. Just as you can have two finite things be of different sizes, you can have two semi-infinite things be of different sizes too. Consider the following:
Semi-infinite library C has an endless number of copies of the Bible. It contains no other books.
If a finite set of books is added to this semi-infinite library, then this library will increase in the total number of books it contains but the semi-infinite number of books it has will not increase. The increase in its number of books must be expressed as semi-inf x + finite x. If another semi-infinite set of books is added to this library (let’s say an endless number of copies of the Quran), then the semi-infinite number of books will have increased such that the semi-infinite number of books in the library is now twice as big. A more comprehensive or bigger semi-infinite you might say.
The biggest possible semi-infinite library R is defined as follows: That which is closest to literally containing all of x, is the biggest library.
Let’s say R contains all books duplicated and original, except ‘10’. Since it is missing book 10, it is classified as incompletely containing x as opposed to completely and infinitely containing x. Where what has been defined is not absurd, this library is the closest thing to infinitely/completely/truly containing x. It matches x 99.99999…ad infinitum%. There can be no infinite library because no library can 100% contain x or be Infinity/Omnipresent. Where R contains all books except ‘10’ and ‘9’, then R does not match x 99.99999…ad infinitum%. The best that we can describe it is that it matches x 99.99999…ad infinitum% minus one book, or, we can describe it as matching x 100% minus 2 books. Admittedly R seems absurd. You cannot take away from the infinite. You can describe parts of it. What is the biggest part of it that you can describe that is not the whole of it?
You cannot have two xs because x encompasses all books (both original and duplicated). Where R is literally x minus one book, you cannot have more than one R. Existence is such that it contains x and R. R contains almost all of x (which means Existence also contains almost all of x because R is in Existence) but Existence contains one more book than R. Thus, Existence contains x, and the manner of its containing x is such that all but one book is in R (which is also in Existence, thus it is in Existence) and one book is not in R (this one book is also in Existence, thus it is in Existence). Both x and R are members of Existence and not members of themselves. Existence is a member of Itself.
Where R is not absurd, it should be described as a super semi-infinite. I’m almost certain (though not yet fully certain) that there can be no super semi-infinites. But there clearly can be different semi-infinite sizes, and Existence clearly is Infinite/Omnipresent.
Solution to Cantor’s paradox:
E = the set of all sets. E is a member of itself. This is literally describing Actual infinity. This is literally describing Existence. True/actual infinity contains infinitesimal. Infinity and infinitesimal refer to the exact same thing just looked at from a different angle/perspective. Existence is infinite through and through. In relation to us, infinitesimal is the internal aspect of Existence and infinity is the external aspect of Existence. We zoom into ourselves past atomic level, there will be something else ad infinitum (rejecting this implies that Existence has an internal end…which is clearly absurd). We zoom out of ourselves past planet level, there will be something else ad infinitum (rejecting this implies that Existence has an external end…which is clearly absurd). An Existence that is not actually/truly infinite is blatantly absurd as it implies the existence of non-existence or something coming from nothing (which is rooted in the absurdity of the existence of non-existence). A temporally finite existence implies that non-existence existed (absurd) and existence came from it. A spatially finite existence implies that it is surrounded by non-existence. This is absurd as it logically implies that non-existence exists and it is surrounding existence. For us to zoom in to ourselves and find and end (the rejection of infinitesimal) logically implies the rejection of Actually infinity. This is clearly absurd.
A clear distinction must be made between actual infinity and potential infinity. Potential infinity is not truly infinite like actual infinity. So it should not be described as being infinite at all. Actually infinity makes the potentially ‘infinite’ possible. Cantor appeared to have thought that everything is just potential ‘infinity’. He did not seem to recognise or understand true/actual infinity. If he did, he would not say there is no set of all cardinalities. Clearly, the set of all cardinalities is Existence/Actual infinity. Again, actual infinity makes potential ‘infinity’ possible. How clearer can it get in terms of what contains the set of all cardinalities?
What contains infinity? Infinity (and there are no different sizes of infinities)
What contains semi-infinities? Infinity (and there are different sizes of semi-infinities)