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Can we conceptually build the (mereological) “sum” of “all being” / “everything that exists” without running into problems or paradoxes?

Can we think of the resulting concept (sometimes called “Being”, capitalized) as referring to an existing thing itself – which means it also belongs to “everything that exists”?

Because for typical, restricted collections this is not the case. The “sum of all cats” is not a cat itself.

It works for mass nouns like “water” (“the sum of all water” is itself water). Or in case of rather unnatural constructions like “the parts of the Eiffel tower” (assuming we regard the Eiffel tower as a part of itself) – which we can ignore since they start from the whole object as given.

So why does it work for “being”? Why is it so obvious, without any justification, that “being” behaves completely unproblematically according to ordinary mass noun “logic”?

Are there any critics of the concept of the “sum of all being” as an existing thing itself?

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    Clearly, "the world” is not the totality of "everything" that "exists" because people talk about many worlds and typically do not include sets, numbers and the like. So both "everything" and "exists" are qualified. The latter typically to physical existence only, or less than that, like the observable universe, so the Cantorian problem does not arise at all. The sentence does not carry the burden of defining, it is just a pointer to whatever is physically out there. However, see Williamson's Everything on defending unqualified use of "everything".
    – Conifold
    Aug 26 at 19:56
  • @Conifold I took it from Wikipedia In its most general sense, the term "world" refers to the totality of entities, to the whole of reality or to everything that was, is and will be. So what would be the correct term? “Being”? “Reality”? All those terms have different nuances of meaning…
    – viuser
    Aug 26 at 20:03
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    There is no "correct" term (i.e. one that faces this problem). All of those are implicitly qualified and/or non-definitional. This problem is specific to mathematical terms that are "defined into existence" with only explicit qualifications, if any.
    – Conifold
    Aug 26 at 20:07
  • @Conifold I didn’t claim it faced that problem – just something roughly analogous. “The world” is sometimes described as “a mereological fusion of everything in it” (Achille Varzi). Or sometimes as the totality of contingent truths. Anyway, we pack a lot into this “world” (and seemingly even more into “Being” or “reality”), so it’s a bit surprising that such constructions yield no inconsistencies.
    – viuser
    Aug 26 at 20:34
  • Well, we need no type distinctions and explosion of classical logic to get to the paradox. Neither is a feature of natural reasoning. Even mild type stratification, as in Quine's NF, provides a non-paradoxical universal set. Williamson sketches how to manage "absolute everything" in a non-paradoxical fashion too, based on higher order logic, which is also no stranger to natural reasoning. So it is not that surprising. Indications are that the paradox is an artifact of particularly rigid and narrow logical devices.
    – Conifold
    Aug 27 at 4:36

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