Mereological Foundation of Set Theory
MAIN RESULT:
The axioms of set theory can be seen to be grounded in the attempt to have a maximally comprehendible representational object world of a predicate hierarchy
The system suggested here to ground that attempt is the simplest possible (mereo-representationally speaking) of such worlds.
DETAILS
The theory I'm speaking about here is Reflective Atomic Representational Mereology. which I've coined lately. It can be shown that Ackermann's class theory is interpretable in this theory, and thus ZFC also. However, the primitives are more natural here. However, there are some technical issues that demand some clarification regarding the natural background of the scheme of reflection, which seems more of a technical powerful tool rather than being an intuitive naturally motivated axiomatization, all the other axioms of Atomic Representational Mereology do capture a specific analytic piece of thought.
EXPOSITION
Language: First order logic
Primitives:
Identity "=", where x=y is read as x is identical to y
Part-hood "P", where x P y is read as x is a part of y Representation"Rp", where x Rp y is read as x represents y
Axioms: those of General Extensional Atomic Mereology (without bottom) +
Note: The General composition principle can be phrased in terms of atomic part-hood as:
General Atomic Composition Principle : if phi(y) is a formula in which x is not free, then all closures of (Exists y (atom(y) ^ phi(y)) -> Exists x for all y (y atom of x <-> atom(y) ^ phi(y))) are axioms.
Where 'atom of' is defined as: y atom of x <-> atom(y) ^ y P x
Axioms of Theory of representation of extensions which are:
[1] a Rp x ^ b Rp y -> [a=b <-> x=y]
[2] Exists! x: atom(x) ^ ~Exist y (x Rp y)
Define: x=0 <-> atom(x) ^ ~Exist y (x Rp y)
[3] a Rp x -> atom(a)
Define (extension) : extension(x) <-> x=x
so every object is an extension.
Define (set): set(x) <-> x=0 or Exists y (x Rp y)
Define (element): x element of y <-> atom(x) ^ x P y
Define (member): x member of y <-> Exists z (y Rp z ^ x element of z)
membership is symbolized by "E", while element-hood symbolized by "e"
A formula phi is a formula of the LANGUAGE OF SETs if and only if the only non-logical symbols appearing in it are those of identity and membership.
An extension is said to be representable if and only if there exists an atom that represents it.
Define (representable): Representable(x) <-> Exists y (y Rp x)
Definition: An extension is said to be completely representable if and only if every sub-extension (i.e. part) of it is representable.
Definition: A set is said to be separatable if and only if the extension it represents is completely representable.
Definition: A set is said to be Hereditarily separatable if and only if it is separatable and every element of the transitive closure extension of it is separatable.
Definition: A set is well-founded if every sub-extension of the extension it represents has an element that doesn't have a member that is an element of that sub-extension.
Definition: A principal set is a well-founded Hereditarily separatable set.
Definition: The transitive closure extension of a set x is defined is the intersectional extension of all transitive extensions of which the extension represented by x is a sub-extension.
Definition: a transitive extension is an extension in which every member of an element of it is an element of it.
Set construction axioms:
[1] Axiom Schema of Reflection: if phi(y) is a formula in the language of sets, in which only y,z1,..,zn occur free, then
for all z1,..,zn are principal sets
[for all y (phi(y) -> y is a principal set)
->
Exists x (x is a principal set ^ for all y (y E x <-> phi(y)))]
is an axiom.
In English, if phi is a predicate expressible in the language of sets, that is closed on the principal set world, then phi is represented by a set in that world.
/ Theory definition finished.
In some sense, this theory offers some explanation of Ackermann's vague set-hood primitive predicate. Here this is explained as being a well-founded Hereditarily separatable set. The ideas behind all axioms except the last are pretty much natural and very clear. However, the last schema of reflection does need some hard justification to prove naturality.
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The main problem is how to naturally motivate the last axiom schema? what that scheme is saying is that if a property that is definable in the language of set theory (from principal set parameters) can be completely realized in the object world as an extension of principal sets fulfilling that property, then this means that this property is realized as a part of the world (the total extension) of principal sets (we may call that world as the principal set world) and thus its atomic "representation" is also to be a part of that world! While on the other hand, properties that cannot be completely realized as a part of the principal set world (like being an ordinal) would not have atomic representations being a part of that world. This is a form of "parallelism" concept where the representation follows the represented. There are actually two aspects involved in this line of thought, that the completeness of realization of a predicate as an extension that is a part of the principal world would mean that a representative of that extension (and thus of the predicate) would exist, and the other aspect is that this representative must also be a part of the same world of what it represents. A natural background that motivates this can be seen in the Parallelism of the object world to a hierarchy of predicates. According to this parallelism, we desire to illustrate a Predicate hierarchy in the Object world. So we want a REPRESENTATIONAL OBJECT WORLD of a PREDICATE HIERARCHY. Now a predicate can be REALISED as an extension of objects in that representational world, for example, the predicate circle can be realized as the extension of all circles in that world. But this is not enough if we also desire to objectively represent having the predicate circle itself being predicated by a higher predicate that doesn't predicate the individual circles, like for example by the predicate shape, so to have that we must have a representative of the predicate circle itself that doesn't have the individual circles being parts of it. Now this object that stands for the predicate when it is fulfilled by a predicate, must fulfill properties that qualify it to be an element of another extension that is part of that representational world for that world to parallel the fulfillment of predicates in predicates. So to COMPLETLY represent a predicate in the representational object world we need that world to grasp two aspects of representation, the first is the extension of ALL objects that fulfill that predicate, the second is the atomic representative of that extension and thus of the predicate, the first represents the predication role of the predicate, while the second represents the predicate when being predicated upon by a predicate, the first is to be called "Representation of a predicate by an extension", the second to be called "Representation of a predicate by a set". If a predicate lacks one of these representations then it is not completely represented in that world even though it might be completely realized in this world as an extension of all of what fulfills it. What we desire from a representational object world of a hierarchy of predicates, is for it to completely represent predicates that are completely realized as extensions in it, and that's what the reflection axiom schema is essentially saying. Now the principal set world defined above is the simplest* possible representative mereological world from the viewpoint of representational Mereology. Here in this method, the principal set world is conceived as enjoying the qualities that a representational object world of a predicate hierarchy must meet! So, in other words, this method mounts to speaking of the Simplest possible representational Mereological object world of a Predicate Hierarchy. This in some sense might conceptually ground this approach. However, practically speaking there is no doubt that here it is the technical aspect that is the immediate motivating impetus. Technically speaking the Hereditarily Separatable sets are maximally comprehendible sets from the viewpoint of comprehending predicates in terms of sets and classes, and having the well-founded structure, this will even make them the most precious kind of sets possible, hence their naming here as "principal", so definitely the reflection scheme is a practical "consideration". The reflection scheme is nothing but the attempt to carry this maximal comprehension further up the hierarchy so that we can maximally comprehend more and more properties about those sets, and so on. Since there is no obvious inconsistency with that, then the possibility of it being consistent (let alone being TRUE) constitutes without any doubt a justification for investigating it. So here we are talking about the Simplest Possible Representational object world of a Predicate Hierarchy that is at the same time maximally comprehendible!
So, in nutshell, all axioms of set theory (ZF, NBG, MK, Ackermann) can be seen to be grounded in the endeavor of having the simplest (mereo-representationally speaking) maximally comprehendible representational object world of a Predicate Hierarchy that can be spoken in the language of set theory. So this supplies a sort of Analytic theme that the axioms of set theory can be seen to be about!
Now the restriction to the language of sets, though being technical, yet it is naturally expected since after all we want to provide a foundation for set theory, and predicates that matter to set theory are those defined in its language. Of course to be in the language of set theory then it must not quantify or use parameters that can range over non-set objects, however the further restriction made here to the principal sets in parameters (although I think it is not necessary technically speaking, yet it is necessary for naturality of this theory), confines to the idea of speaking about predicates in the language of sets that are about principal sets. This closure property of predicates within the principal set world, without doubt, conforms more to the general purpose of defining the representational object world of principal sets mentioned above. All of this without any doubt give some conceptual flavor to this method besides its obvious pragmatic stance.
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*Simplist is meant here to be in the Mereological and the Representational sense since atomic Mereology is far simpler than gunk Mereology and preventing multiple representatives of the same extension is a simpler representation than allowing piling up of copies of them, so it conserves both material and representation.
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Addendum: 1/10/2017
To interpret Ackermann's class theory, we take the membership relation E* defined as:
x E* y <-> x E y or (~set(y) ^ ~Exist z( z Rp y) ^ x e y)
The domain D will be taken to range over all sets and
non-representable extensions.
Now all of Ackermann's axioms written with epsilon membership replaced by E* and = by =, and all quantifiers restricted to D, will be proven in this theory.
Addendum 2020.
This development needs to add an axiom that:
for any formula phi the extension of all principal sets that satisfy phi is representable.
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A more recent equivalent system is presented here: