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The suprising thing to me is that this attempt have a simply stated single schema on top of the axioms of Labeled Mereology, that can interpet all of Ackermann's set theory and therefore interpret ZFC. Simply take Ackermann's mystereous set predicate to be hereditarily pureset and all axioms of his set theory would be go through under that interpretation.

The suprising thing to me is that this attempt have a simply stated single schema on top of the axioms of Labeled Mereology, that can interpet all of Ackermann's set theory and therefore interpret ZFC. Simply take Ackermann's mystereous set predicate to be hereditarily pureset and all axioms of his set theory would go through under that interpretation.

A part from the question of legitmacy of having a Bottom atom in Mereology (which can be done without it here, but the formalization becomes more complex), did the definitions given here adequatly capture the true nature of the notions they define? I mean as relates to those notions as treated in modern standard formal set theories which usually do not use Ur-elements.

A part from the question of legitimacy of having a Bottom atom in Mereology (which can be done without it here, but the formalization becomes more complex), did the definitions given here adequatly capture the true nature of the notions they define? I mean as relates to those notions as treated in modern standard formal set theories which usually do not use Ur-elements.

The point here is that this axiom does depend in an essential manner on the defined notions of sets, classes and their membership; it essentially depends on the internal make-up of sets, I mean in Mereological terms! That is, what they are composed of. So it features Ackermann's sub-world $V$ as the world of those precious kind of sets, the most pure form that a set can be of, i.e. those sets which were described by Lewis. On the other hands proper classes are classes that are not Lewisian-sets. This is a tempting explanation of Ackermann's set world.

The point here is that this axiom does depend in an essential manner on the defined notions of sets, classes and their membership; it essentially depends on the internal make-up of sets, I mean in Mereological terms! That is, what they are composed of. So it features Ackermann's sub-world $V$ as the world of the most precious kind of sets, the most pure form that a set can be of, i.e. those sets which were described by Lewis. On the other hand, proper classes are classes that are not Lewisian-sets. This is a tempting explanation of Ackermann's set world.