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As for the domain, I wonder,
If truly nothing existed,
would the non manifested potential to exist remain ?
In physics, the Lamb shift, named after Willis Lamb (1913–2008), is a small difference in energy between two energy levels 2S1/2 and 2P1/2 (in term symbol notation) of the hydrogen atom in quantum electrodynamics (QED). According to the Dirac equation, the 2S1/2 and 2P1/2 orbitals should have the same energies. However, the interaction between the electron and the vacuum (which is not accounted for by the Dirac equation) causes a tiny energy shift on 2S1/2. Lamb and Robert Retherford measured this shift in 1947, and this measurement provided the stimulus for renormalization theory to handle the divergences. It was the harbinger of modern quantum electrodynamics developed by Julian Schwinger, Richard Feynman, Ernst Stueckelberg and Shinichiro Tomonaga. Lamb won the Nobel Prize in Physics in 1955 for his discoveries related to the Lamb shift.
Countably infinite is the idea that I can "count" from 1 to 2 to 3 and eventually work my way up to infinity using the set of integers (Z).
In your example the 0 represents a place holder for a potential object
Ok I get what you mean with the sets, but am I wrong in claiming that the 0 still represents 0? ... , but if it symbolizes 0 amount, then 0 is still 0. If I had 0 apples in my fridge, and owed you 3 apples, I would have -3 apples in that set perhaps? If I got 3 apples to give to you, then I would have 0 apples.
Ok I get what you mean with the sets, but am I wrong in claiming that the 0 still represents 0?
If I was born a Swede (jus soli) and had a parent who was a US citizen (jus sanguinis). Then I am both a Swede and a US citizen. However at any given moment I can choose to reside in Sweden or the US due to the dual citizenship (i.e. being a ∧ b allows for a ⊕ b). If I am neither Swedish nor a US citizen (¬(a ∨ b)) I wouldn't be allowed to permanently live in either country. So similarly if 0 is neither positive nor negative (¬(+ ∨ -)) it can never be +0 or -0 because it isn't of the same category. Meaning taking 0 as ¬(+ ∨ -) for granted as true, would indicate a + 0 ≠ a because unary +0 by its very definition would be invalid. As a consequence of this, since 0 acts a placeholder for positive and negative quantities across all number bases. It naturally follows that 0 is both (+ ∧ -).
I know you say eventually work way up to infinity, but I would say it is not impossible to work your way up to infinity, but it is impossible to arrive at infinity. There are infinite amount of numbers that will never be written, actually infinite times infinite, actually infinite times infinite times infinite, actually...
Exactly, very good. That is why the inverse of zero is *any* complex number. It has the potential to be anything. This should start to give you an idea of why zero is connected to infinity.
At the center of the space of all "absence" is something that very nearly mimics infinity. How? From a+b=a-b we can deduce: b = 0 = (a-a)/2 ; So 2 = (a-a)/b ; Thus, 0 ≈ (a-a)/(a-a/(a-a/(...))) ≈ 2.
Since, personally, I strongly suspect the Mathematical Universe Hypothesis (discover article) has validity. I believe there is an infinite intangible number space (something that doesn't require physical space, like, well, numbers) contained in absence because absence is the outer boundary of an infinite centric number line (infinity → 0 versus our reality of 0 → infinity).
In the French convention, zero is considered to be both positive and negative. The French words positif and négatif mean the same as English "positive or zero" and "negative or zero" respectively.
The paper makes the argument that 0, as the additive identity element, must be positive and negative for four reasons.
So what are the ramifications of all this?
The paper shows that by using these characteristics, just like how we can write:
 +  = [0,3]
0 + 3 = 3
That we can potentially convert a set  (neither + nor -), to a quantity like 0 (as + and -), or even to a contradiction Ø ≠ Ø (none of the above). The more interesting possibility though is to get closer to the center infinite-like characteristic. Which should partially address ...
Ein Sof is the divine origin of all created existence, in contrast to the Ein (or Ayn), which is infinite no-thingness ....
It is the origin of the Ohr Ein Sof, the "Infinite Light" of paradoxical divine self-knowledge, nullified within the Ein Sof prior to Creation. In Lurianic Kabbalah, the first act of Creation,....
.... giving rise to Monistic Panentheism.