This is an ontological engineering question, please treat it that way.

https://en.wikipedia.org/wiki/Ontology_engineering I am examining this question from the point of view of ontological engineering. Russell's paradox is based on forming a set of all sets that do not contain themselves.

Within the last few years it occurred to me that set membership could be defined through ontological engineering to inherit from a generic base class: total_containment. Both physical and conceptual total containment would inherit from the base class.

In order for a thing to totally contain another thing the outer boundary of the contained thing must be entirely contained within the outer boundary of the container. This forces the container to be somewhat larger than the contained and thus makes it impossible for anything to totally contain itself.

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    Questions of engineered ontologies and their implementation on computers should be asked in AI SE. ai.stackexchange.com – J D Nov 21 '19 at 17:35
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    No, there are versions of set theory that allow such sets, see non-well-founded set theory, e.g. Aczel corresponds sets to accessible pointed directed graphs. – Conifold Nov 21 '19 at 18:50
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    @user4894 This question is pushing the OP's personal viewpoint (see also the comments below my answer where the OP argues that my answer, which develops the quite correct point in your comment, is irrelevant to their question). That already makes it opinion based; one might decide to give the question the benefit of the doubt (at least temporarily) but previous interactions with the OP have convinced me that there is no reason to do so. (Note that I do agree with you that the title question is perfectly fine; when I initially gave this question the BotD it was that that I was responding to.) – Noah Schweber Nov 21 '19 at 19:54
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    @PL_OLCOTT I'd suggest you are conceptualzing knowledge with the containment metaphor, and that an analysis of the nature of propositional knowledge would show that the metaphor is a limit. Propositions can be inherently recursive, and that this violates the extensionality of space shows the limitation of mapping an ontological primitive of "meaning bearer" to an ontological primitive of physical space such as "container". en.wikipedia.org/wiki/Conceptual_metaphor – J D Nov 22 '19 at 0:17
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    You're confusing figurative and conceptual metaphors. Concepts don't literally contain anything anymore than memory cells are actual cells in a building. Neither are literally containers. Here's a construct that contains itself. en.wikipedia.org/wiki/Quine_(computing) – J D Nov 22 '19 at 19:09

EDIT: it may be worthwhile for those interested in answering this question to familiarize themselves with the OP's earlier posts throughout the stackexchange site - e.g. here (now deleted), here, or here - under this and related usernames. I did not check the username ahead of time, or I would not have answered; that said, I've decided to leave this answer up since it may be useful for other readers.

Re: my decision to vote to close after answering while not deleting my own answer, see this meta discussion.

No, it's not.

(Unless you interpret "logically incoherent" with "incompatible with my own intuitions about sets," which is - to put it mildly - questionable).

First of all, the claim that regularity in ZF resolves Russell's paradox is a (very common) mistake. Adding an axiom never removes inconsistencies; any contradiction present in the weaker system remains present in the stronger system. Rather, what saves ZF from Russell's paradox is the removal of full comprehension - mere separation only lets us prove that for every set A the set R_A:={x in A: x not in x} exists, but that's no paradox at all (the Russell argument shows that R_A is not an element of A, but so what?).

What the axiom of regularity does is imply that the class of non-self-containing sets is the entire universe. But that's a separate issue.

More substantively, there are set theories which permit (indeed, require) self-containing sets which are known to be consistent relative to theories we have high degrees of faith in. For example, the theory ZFC - Regularity + Aczel's antifoundation axiom is consistent if ZFC is, and proves the existence of self-containing sets.

For more dramatic examples see e.g. this article of Holmes, especially section 6.2 - the point is that there are set theories which imply the existence of self-containing sets with extremely weak consistency strength.

And in a more normative direction, various authors have actively argued for such theories; it may be worth reading their arguments (see e.g. the sources in the Stanford Encyclopedia article).

All that you've argued is that self-containing sets contradict one particular intuition about sets. But nobody has a monopoly on that notion. Unless you interpret "logically incoherent" as "in contrast with my own beliefs," there's no issue here.

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  • What I am proposing logically follows form the semantic base meaning of the term "total containment". Within the goal of deriving the natural pre-existing ontology for the set of all knowledge we are not free to overload one concept with another incompatible concept. We can at most further elaborate the base concept by providing more details. – PL_OLCOTT Nov 21 '19 at 16:53
  • When we take the common meaning of {total containment} as our base concept we can only add and not override its base meaning. When its base meaning stipulates any totally contained object must have its outer boundary inside of the containers outer boundary then anything totally containing itself becomes a logical impossibility. – PL_OLCOTT Nov 21 '19 at 16:53
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    @PL_OLCOTT You asked "Is a set that contains itself always logically incoherent?," not "Is a set that contains itself incompatible with the following approach to sets?" If you wanted to ask the second question, then ask it; otherwise, the answer above is correct. – Noah Schweber Nov 21 '19 at 16:54
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    @PL_OLCOTT Haha, I answered this question without checking the username. I have no interest in engaging with your nonsense. I'm leaving this answer up since it may be useful for other readers (your frivolous downvote notwithstanding) but I will not respond further to you. – Noah Schweber Nov 21 '19 at 16:56
  • I was asking the question from an ontological engineering point of view. I did not make this quite clear enough initially. Your answer was clearly not from an ontological engineering point of view. When we look at things from an ontological engineering point of view it is a matter of how concepts are constructed from their constituent parts. When we try to construct the concept of a set containing itself on the basis of the generic concept of total containment the pieces do not seem to fit together. – PL_OLCOTT Nov 21 '19 at 20:13

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