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I would have to say that ultimately Everything encompases all that exists as well as all that does not exist. Perhaps my logic is flawed with the initial assumption that A={Non-existence}={}? Is this set truly empty?
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I believe A should indeed not be empty.
Keeping your variables for clarity if we agree that {non existence} is different than {nothing}:
a = { Non existence} = { All that does not exist }
b = { Existence } = { All that does/has/will indeed exist }
c = { Everything }
d = { Nothing } = { ø }
If we do agree that everything is not simply what exists ( c != b )
but rather that everything encompasses all that exists as well as all that doesn't:
( c = { a U b } )
what about d? Does "nothing" exist? Isn't it true that it both does and doesn't? +0 is the same as -0...
a = {Non existence} = { c - b + d} = { c - b - d }
In the truest sense of the word, everything is the opposite of nothing, therefore nothing should be excluded from everything. But there is a duality to that.
Does excluding nothing from everything mean that:
c = { a U b }?
Or that nothing is excluded therefore all variables are part of {everything}, including nothing...
c = { a U b U d }?
I think that both are true. Therefore everything both contains and excludes nothing.
Thus my definations of everything and nothing are:
Everything is all that is and all which is not, all there ever was and all which never was, all that will ever be and all which will never be.
Nothing is what is not.
Kind of strange how time seems to have no meaning in the defination of nothing. I could not say that nothing is what never was can I? I used to have a sandwich and now I have none. I can't say nothing is what will never be either, cause I'm making another.
Time is an even stranger concept in our defination of non-existence. Sure it's what does not exist, but it's not what will never exist. My son does not yet exist but someday I might have one. Or maybe I had one and he died... So if I had one in the past he goes in column a (exists), if I will have one in a few years he's in column b till then. Since I dont know if I will ever have a son does my son exist or not? I'd say he doesn't but existence was defined as all that ever will exist... So by that defination my son neither exists or doesn't till the die I'm dead or infertile.
So if we relook at existence, shouldn't it be what DOES exist, as in NOW, not tomorrow or yesterday. If so, everything then should be what exists presently as well, not what will or may.
By that logic, everything's defination depends on your perception of wheter time is indeed linear, and wether events in the future are predetermined or not.