I would not prefer to describe any attribute for God as necessary in any sort of sense other than for convenience in conversation, because humans simply haven't the authority, and one doesn't want to trick oneself.
That God is omnipresent or omniscient still presents difficult to define concepts (see Plotinus) especially since the nature of the Universe and of human experience is uncertain. Even so, to prove that God does not exist one should need to comprehend existence fully and completely search through it-logically. Perhaps that point is difficult to make without an example; I will write one in a minute or two.
A verse-even a uni-verse may have several definitions and boundary conditions itself that are conventions. Cardinals of trans-finite series (you are correct about Cantor's term) that are variable in reality as well as in theory representing time values for spatial size appear malleable. The solid state Universe description is a description of human experience of mass-energy of a portion of what may be a very large Universe from an inflation or whatever event occurred during the Plank epoch.
Plotinus described the One as unextended generally. The Intelligence issued the realm of forms and the solid state Universe with its many broken forms (something like broken symmetry in a physical cosmology perhaps). It is very difficult to say anything that God must do as a necessity in His constructions because of the uncertainty of knowledge of the relationship. Logical investigations are quite useful for insight into the unknown, yet logic is for-itself a phenomenon of human reason without absolute grounding in anything besides the circumstance of being.
Karl Popper said that one can't define rationality-one just recognizes it. I believe that criteria applies to God as well; His grace is entirely agape and disclosure of teleos at soley with His will.
Perhaps a trans-infinite series of numbering might be created for infinite series with a variable cardinality or for comparing the absolute value of infinite series regarding relative location. Well, I am not a mathematician.
An example of why it is important to actually look through things sometimes to confirm a hypothesis rather than leaving it to theory.
A Philosophical Discussion Aboard the First Ship Arriving In The New World
‘Well my scientific advisor, what think ye on our prospects for finding natives in the new continent?
‘That be impossible captain. No one knows what a New World native is therefore they cannot exist.”
“Well, we have a general idea of what they might be like even if not specifically.”
‘Such be logical incoherence mon Captain amigo. There be no such thing as N.W. natives.
“Would you like to swim ashore to look my friend, in order to confirm your theory?
‘Land be not in sight yet Colin’
‘’Ahem’
‘Let me clarify the point. N.W. natives will be some kind of generic human being with unspecified features native to New Worlders living here for thousands of years without contact with Madrid, Barcelona or Paris.”
‘Can’t be anyone lives in the New World. No one knows what they would be like’
“Mr. Science, you would need to search every bit of the continent to prove that is has no population of natives.”
“Why no my glorious Captain, you would need to prove that natives exist with logic in order to contradict my opinion that they don’t and cannot exist.”
“Mr. Science, I am asking your scientific opinion.”
‘Yes sir; my opinion is that N.W. natives cannot exist because they are not defined.
“Good God Mr. Science!”
“God cannot be defined and therefore cannot exist either my Captain.”
“Nothing can exist without a good, complete definition then?”
“That be exact and logical sir.”
“Look Mr. Science, a new continent we have never before seen is dead ahead.”
The Captain ordered the executive officer to slow the ship to let natives in canoes approach..
‘Mr. Science, I shall award you an ostrich feather for your hat. You have invented the necessary equation for de trop existence. The Mr. Science colonialist formula;
For every X, X can exist if and only if Y can define X.
If Y can define X, then X can exist.
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