The Essence of Sets

Sets are dynamic mereotopological packages.

[Note (written at 27/1/2018), I think of all my thoughts about what sets are, this one is the nearest to the essence of what sets are!: However I'll add an informal definition to support the above quasi-technical definition, I'd say that:

A set is the form of distribution of material of an object in terms of discreteness and continuity. 

In other words, sets are packages of the material of objects in terms of continuity and discreteness]

Introduction

This is my own personal understanding of what sets are. It extends Dynamic Mereo-Topology (discipline devoted to understanding relations of part-hood and timed external contact (touch)). Variation in contact between objects over time makes it possible for the same object to have different packaging of its material over time, that's why packaging is added as a primitive two place function in order to address this variability. So an apple for example can have different packaging at different moments of time, at some moment it is packaged as a "unit", i.e. every proper part of it is in touch with a proper part of it and the apple itself is not in touch with any object, so it is not the union of two non contacting parts, i.e. there is one whole apple, at other times it can be split into two non contacting parts, at other times into n many non contacting parts, etc...  However, in all of these states it is the SAME apple mereologically speaking; those different states are what is referred to here as "Packages" of that apple. It is the same apple because no extra-material was added to it nor any material was removed from it, but the packaging of its material had differed. The following account will attempt to understand sets along that vein. So sets are defined as packages here, membership in them is defined as being a "unit" part of the packaged object at the moment of packaging, and rules capturing naive properties of packages are laid down. It would be seen that naive thought about packaging would indeed interpret all axioms of Zermelo Frankel set theory minus infinity, and even infinity itself would be seen to be proved in a rather trivial offshoot of this line of reasoning and that this in some way provides a justification of infinity.  The key elements of this line reasoning lies in David Lewis's aesthetic principle of  "subsets are parts of sets", and in providing an explanation of his Singleton function through reasoning of packaging of a unit as being an atomic process and thereby constituting a mereological atom, and in specifying conditions when to expect objects to be packaged as a unit, those conditions would follow the most naive thought that strikes ones attention when experiencing that method, that of a size constrain, and the last pivotal concept is that a very simple variant that size constrain prove infinity! Thereby proving all axioms of ZF set theory except Regularity.

Background

The mereological background of this theory is simple, that of General Extensional Atomic Mereology with Bottom. Then we add to it the timed contact relation and the packaging function.

General Extensional Atomic Mereology with Bottom is well known, I'll proceed by giving an informal account of it with some detail formulation of some key aspects and supply a link for detail treatment.

Informally the relation "is a part of" can be understood as "having its constituting material being constituting material of", so, for example, the nose of Mr.Simpson is part of the head of Mr.Simpson, all of constituting material of the nose of Mr.Simpson are also constituting material of Mr.Simpson's head, while this is not the case for Mr.Simpson's leg for example and that's why Mr.Simpson's nose is not a part of his leg. So accordingly one can understand the basis of the following statements:

1. Every object is a part of itself.

2. A part of a part of an object is a part of that object.

3. No distinct objects can be part of each other.

Actually, we can DEFINE identity of objects after having the same parts, i.e. every part of either is a part of the other.

Now we come to understand what is an atom. Ideally, an atom is ought to be an object that has no proper parts, i.e. has no parts that are distinct of it. But since we are adopting here an approach to mereology in which a bottom object "o" exists, i.e. an object that is a part of every object, then this would render the bottom object o being the sole atom in the above sense. So we resort to another definition of mereological atom that of:

Define "x is an atom" if and only if every part of x other than o is x itself.

This violates one of the well recognized mereological principles that of strong supplementation which states that for any proper part of an object there must exist a proper part of that object that is disjoint (doesn't share any part) of it, here this is violated since nothing can be disjoint of o, but nevertheless all rules can be easily changed as to allow for existence of a bottom object, even strong supplementation itself can be modified. Now we'll restrict ourselves to the field of Atomic Mereology, i.e. to objects that are constituted of atoms, in other words, we do not allow for the existence of atom-less objects or the so-called gunk objects, which are objects that are infinitely divisible, i.e. has no atom among their parts. So to effect that we add the following rule:

4. Every object has a part of it that is an atom.

Also to accommodate for adding the bottom object we need to modify the

definition of disjoint to from the usual notion of not sharing any part

to that of not sharing a part other than o since all objects do share o by definition.

The last two principles of Mereology are the supplementation and the Composition principles. The first is a modification of the usual one by adding the condition of "not being bottom", so if an object has a proper part other than bottom then it must have another non-bottom proper part that is disjoint of the former in the above sense. The composition principle is the well known unrestricted composition principle of mereology. A simple variant of it is presented here which simply states that for any property phi that holds of at least one atom, there exists a mereological totality of all atoms fulfilling phi. This is:

5. if phi is a formula in which x doesn't occur free, then all closures of 

[Exist an atom k (phi(k))] -> Exist x for all y (y P x ^ atom(y) <-> atom(y) ^ phi)

are axioms.

where P denotes "is part of".

For a more thorough treatment of those mereological notions, one can refer to Varzi at:

http://plato.stanford.edu/entries/mereology/

For now, it is important to get into the definition of "is a unit", this requires

understanding the next primitive concept of this method that is "timed

external contact" or simply "timed touch". Before we go to "timed" touch

lets comprehend what "touch" is in the first place. Here touch is just a kind of external contact between two objects the naive rules about it are the following:

1. An object can only be in touch with a disjoint object.

2. If an object touches an object then the later touch the former too.

3. If an object touches a proper part of a disjoint object then it touches that object too.

4. If an object touches an object that is not an atom then it touches a proper part of that object.

Now a unit is an object that is not the union of two non contacting disjoint proper parts, and that is itself not in touch with any object.

Definition: Unit(x) <-> (~Exist k C x) ^ (~Exist yz (y PP x ^ z PP x ^  y disjoint z ^ ~ z C y ^ x= y U z)

where PP signifies "is a proper part of".

where U signifies the union operator which is simply the sum of material in y and z, so x= y U z means x is the mereological totality of all atoms of y and z.

Examples are most of the individual entities that we experience around us can be imagined as "units", for example, an apple, a specific person, a brick, a house, etc.... all of those appear as one totality, they are not the union of two non contacting parts. On the other hands examples of what is not a unit is clearly a multiplicity of many non contacting objects, a herd of cows, a football team, A galaxy, etc.. all those are clearly objects composed of non contacting disjoint proper parts. 

The concept of a unit is primal to this method! Since packages are defined using it and the singleton function of David Lewis is given an explanation here using it. The idea about the complexity of a package essentially depends on it.

Dynamic touch and units

Now we come to visualize matters in a time-dependent manner. Even

in our everyday experience we notice that objects do not always maintain

the same contacts with their environs, and accordingly what is a unit today

might be split tomorrow into a whole of many disjoint pieces, so here time trip in to effect change. Now all of what is said above about touch relation and unit is to be time corrected, and this is just a simple modification, so touch would be a three place relation symbol instead of two, and the third place is occupied by time, it is better written as  x  Ct y  to denote x contact y at moment t, as a consequence the definition of unit would be time dependent, so we have Unitt(x) to mean x is a unit at moment t, this is defined exactly as above but with a timed contact symbol Ct  instead of the symbol C. It is this dynamic change of loculation of material of objects that calls for defining packages of them and thus paving the way to define *sets*.

Packages: 

Generally, package means a form into which material is put, for example a drug package, that can be a tablet, a capsule, or an injection, the tablets can also be of different packages, etc... However Here a package is a primitive two place function, denoted by "y=Packaget(x)" to mean y is the package of x as moment t, package is taken to specifically mean the form into which the material of an object is distributed in terms of discreteness and continuity. For example lets take four atoms and lets denote each one of them by the symbol *.   and lets name them a,b,c,d from left to right. Now the totality of all those atoms is the object that would be denoted as H.

Now lets examine two packages of H, one is the completely discrete package in which all of those atoms are separate (not in contact) from each other, and the opposite package where all of those atoms are in contact with each other, of course, those packages occurs at separate moments of time, but I'll neglect explicit mentioning of time for now.

Package A:    *   *   *   *

Package B:    ****

So package A is the completely discrete package, while package B is the completely continuous package. In package A the object H is composed of four units (other than o) while in package B the object H is composed of one unit. So package A

is distinct from package B although they are packages of the same object H.

I can denote package A as  a,b,c,d  (in any order actually) , while package B

as abcd (in any order), commas serve to indicate non-contact. We can have package C bing  ab,cd and this is:  

Package C:  **  **

This is also different from the prior two, and so on..  Now those different packages, of course, are time-related, so A = Packaget (H), while B is

Packagej (H) and C is packagei (H) where t,i,j are different moments of time of course.

What we need is to have a rule that discriminates between these different packages and also identify identical packages over time. To do that we resort to defining a relation between unit parts of an object at time t and the package at time t of that object. We'll call this relation as "Membershipt", this is defined as:

Definition: x Et y <-> Exist G (x P G ^ Unitt(x) ^ y=Packaget(G))

Now we come to stipulate a rule that determines the identity of those packages:

6. Packagei (A) = Packagej (B) <-> for all x ( x Ei A <-> x Ej B )

Of course, since all objects are totalities of units then clearly two packages are identical only if they are packages of the same object. 

So package here being a two place function would be unique per object and moment of time, and when an object has the same package at different times then this means it exactly have the same form of distribution of its material in terms of discreteness and continuity.

Of course we can define "a package" as a ONE place function as:

p is a package of x <-> Exist t (p=Packaget(x))

So two packages of an object are identical if they share exactly the same unit parts of the packaged objects at the moments of their packaging.

Now we come to define "complexity" of packages, can packages themselves be thought of being more or less complex than each other? and how that can be portrayed?

If we look at package B of object H depicted above, we can notice that there is no much discreteness versus continuity differences involved, H is packaged as a one unit, simply as such, this packaging appears simple and can be imagined of not being composed of proper parts as far as packaging is concerned, in other terms we can stipulate that every package of a unit is a mereological atom!  So package B for the above example would be an atom; while on the other hand packages C and A are more complex. However, can we describe the complexity of those packages, i.e. break them down into smaller packaging processes, the answer is Yes. We can intuitively say that Packaging B, for example, is the union of the individual packages of the units of H, in other words

Package B = Package (a) Union Package(b) Union Package(c) Union Package(b)

Of course, all of those are at the same moment of packaging B. 

This seems to be plausible. Notice that by using the membership relation depicted above then packaging of a unit results in that unit being the SOLE member of that package (at that moment). So packaging of units is the Singleton function of David Lewis. So here if an object is packaged into multiple units then the package of it at that moment is the union of all packaging of units of it at that moment. To apply the above for the example of package C here we'll have:

Package C = Package (ba) Union Package (dc)

Which is plausible. The complexity of a package mirrors the complexity of the object packaged at that time. Now we want to get rid of t in the definition of membership so we define:

x E y <-> Exist t (x Et y). 

One needs a little bit understand that binary relation, here x would be an element of y overall moments of time even beyond that moment that caused the membership in the first place. It is understood that for any object x to become a member of a set then there must be a moment of time where it is packaged as a unit and by then the package of it at that time which would be an atom would be the singleton set of x, i.e. would be {x}. However, this package would remain to be the singleton of x at all moments of time despite the packaging of x at those moments since the above membership relation is not sensitive to what package objects has at other times. 

Now from all of that, we can see nice results that are:

1*)Every part of a package is a package

2*)Every singleton package is an atom

3*)Every package is a totality of singleton packages which are atoms.

4*)The multiplicity of units packaged is equal to the multiplicity of atoms in the package. 

5*)Two packages are identical if and only if they have the same elements.

All of those are timeless statements except 4 which has "units" being time dependent. However, this results in the following timeless expression:

6*) The multiplicity of elements (members) of a package is equal to the multiplicity of atoms that package itself is composed of. 

Now if we define a set as a package, and define subset as a package packaging some of its elements, then clearly we'll have the aesthetic principle of David Lewis that is.

Parts of Sets are Subsets and vice versa.

However the above applies to nonempty packages, but what about the empty package can there be one?

According to this method if we are to stretch Lewis's principle to the most then only the bottom object can be the empty package. 

The kind of packaging discussed in the above example is what can be called as space packaging. This contrasts with another kind of packaging that of time packaging, this happens when we package material over different moments of time, so the total package would be a record of how the material of the packaged object is distributed over time in terms of discreteness and continuity, an example of time packaging is when packaging an object and a part of that object, for example lets take the object H above, and then take the union of Package B and Package (ab), of course, both packages are Atoms since they package objects as units, package B packaged H as ****

and package (ab) packaged ab as **. Now the union of these two atoms would itself be also a package albeit each has a different time of packaging from the other however still it represents a form of repackaging of H together with a part of it over time, this is the time packaging, notice that the resulting union package in the above example has its elements being  H and ab so it is the set { H , ab }. An example of a time package is, of course, the power set. Time packaging is, of course, a stronger concept than space packaging since the later can be seen to be just a single moment time package. So we arrive at the seventh of the nice results about packages, that is:

7*)Every totality of packages is a package.

So this establishes the definition of sets as dynamic (i.e. time-related) packaging. It needs to be asserted that a package is used here not just to mean those denoted by the primitive symbol which is time corrected, which are just spatial packages, no it covers any union of such packages and so includes the temporal packages as well although those are not packages at a specific moment. So packages here are totalities of unit packages despite the moments of those unit packaging.

So although t is referred to here as "moment of time" it is better understood as a totality of moments of time, where each moment of time is an atom of course.

 let's now try to visualize set membership as defined above, by this illustrative example.

What is the set {a,b,c,d}?

The answer would be that there will be an object H which is the totality of all material present in a,b,c,d, i.e. H is the collection of all atoms in objects a,b,c,d, notice that those objects can be atoms or may not be atoms, however H is the union of all of those. Now H is NOT the set {a,b,c,d} no it is the totality of all members of it but it is not that set itself. The set {a,b,c,d} itself is special Package that packages H, that package itself is the union of unit packages, where each unit package packages a part of H as a unit at some moment of time, so the unit package that package part a of H as a unit at some moment of time will be called as "unit-package(a)", and similarly we have a unit-package(b), unit-package(c) and unit-package(d). Now the totality of all those unit-packages is the set {a,b,c,d}. Now the important question that presents itself here is how we are to understand {{a,b,c,d}}? i.e. the singleton set of {a,b,c,d}, i.e. the set whose sole element is the set {a,b,c,d}. Now the union of the above unit packages which is the set {a,b,c,d} is itself not of specific package, its packaging can vary over time, now lets call that union as S, so as said S={a,b,c,d}, now For S to be an element of a set, then S must itself be packaged as a unit at some moment of time, i.e. all those individual unit packages spoken about above must come in contact with each other at some moment say t of time so that S becomes a unit, if that happens at moment t then the package of S at t would be an atom, i.e. we have "Packaget(S) is an atom" Now clearly this atom is the set {{a,b,c,d}} since it has S as its sole member. If S cannot be packaged as a unit at any moment of time then this would mean that S cannot be an element of any set, i.e. S is a proper class. Now formation of sets from singletons is easy since a set is the union of all of its singleton subsets, so x E y is simply having {x} being a part of y which is the same as {x} subset of y. What has been explained here is the Singleton Function itself, something that was found by David Lewis to be shrouded in mystery, while here it is not so shrouded, there is a pretty much naive understanding of it in this methodology that springs about pretty much naive rules about it that readily comes to mind and those rules which simply answer to the question of when do a set have a unit packaging, those rules are simple and not many and even can effect infinity!

The size principles:

Unlike the above principles which are naive and inherent to reasoning about

packages, here the size axioms are 'set' axioms, however, they are also rooted in reasoning about continuity and discreteness in packages, however, those clearly have "collective" aspect thus called as set axioms, and of course they are the axioms that would effect all axioms of ZF except Regularity, so I think this naming that I gave to them suits them well.

The axioms answer to the question of: 

When can an object be packaged into a unit?

The immediate answer that comes to mind is when the number of units in that object is equal to the number of units of an object that is a part of an object that has a unit package at some moment of time.

In other words the ability to bring about all atoms of an object in continuity with each other, thereby packaging it into a unit, depends on the size of that object relative to objects that are known to have such unit packaging, if it has equal or smaller size than those then it can be packaged as a unit!

So, in other words, if there is a formula Phi(x,y) that links units x of object A to units y of object B in a bijective manner, and it is known that A is part of an object C where C has a unit packaging (i.e., there exists a moment of time where C is packaged as a unit), then B has a unit packaging.  Of course, this proves separation by letting phi be the identity function over that part of A.

Now the above principle proves Pairing, Union, Power, Separation, and Replacement.

Extensionality is already a theorem because of the rule of identity of packages.

Then all axioms of ZF-Regularity-Infinity are enacted!

INFINITY

To be noticed is that this theory does prove the existence of an infinite set, but it doesn't prove the membership of that set, i.e. it doesn't prove the existence of a set having one of its members being infinite. The set of all packages is provable from the unrestricted composition principle of this theory, and that is provably infinite. So this theory does prove second-order arithmetic, in which most of mathematics can be grounded. However for those seeking to go beyond that then the membership of an infinite set must be stipulated, this would become a necessity for such a quest. There is nothing in this methodology which points against that possibility, on the contrary, we shall see that this methodology does encourage the existence of such infinite sets!

If we postulate that every bijective function Phi between elements there would be a package in which every pair of elements linked by Phi are in continuity with each other, this postulate has the superficial appearance of being an offshoot of the size criterion depicted above, the only difference is that it doesn't require a unit packaging in advance as the size criterion principle requires, and it is this difference that enables that postulate to PROVE infinity. Take the bijective function that sends each natural number to its successor, apply this postulate and we get the set N of all natural numbers being a unit at some moment!

there are some beautiful related scenarios:

A) Exist t [continuoust(x)] -> Ej [Unitj(x)]

B) X P Y ^ Exist t[continuoust(Y)] -> Exist j [continuousj (X)]

C) If F is bijective from units of A to units of B -> 

Exists t [for every p (p in F -> continuoust(p)) ^

(Exist j (continuousj(A)) -> continuoust(A))] 

Formal Exposition

Language: first order logic

Primitives: 

P(x,y)  for "x is a part of y", infix notation: x P y

C(x,y,t) for "x is in contact with y at moment t", infix notation x Ct y

Pkg (x,y,t) for " x is the package of y at moment t", infix notation: x=Pkgt(y)

Definitions:

x = y <-> for all z (z P x <-> z P y)

x PP y <-> x P y ^ ~ y=x

x O y <-> Exist z (z P x ^ z P y)

bottom (x) <-> for all y (x P y)

x=o <-> for all y (x P y)

atom(x) <-> for all y (y P x ^ ~x=o -> y=x)

x disjoint y <-> for all z(z P x ^ z P y -> z=o)

x= y U z <->  for all k (k O x <-> k O y or k O z)

x= U (y| phi(y)) <-> for all y (y O x <-> Exist z (phi(z) ^ y O z))

Axioms:

1.       For all x (x P x)

2.       For all x,y,z (x P y ^ y P z -> x P z)

       Theorem:  for all x,y (x P y ^ y P x <-> x=y)

3.       For all x Exist y (atom(y) ^ y P x)

4.       For all xy (x PP y -> Exist z (z PP y ^ z disjoint x))

5.       If phi is a formula in which x is not free, then all closures of

Exist y (atom(y) ^ phi(y)) -> Exist x for all y (atom(y) ^ y P x <-> atom(y) ^ phi)

are axioms.

6.       For all x,y,t (x Ct y -> y Ct x)

7.       For all x,y,t (x Ct y -> x disjoint y)

8.       For all x,y (x Ct y ^ ~ atom(y) -> Exist z (z PP y ^ x Ct z))

9.       For all x,y,z (x disjoint y ^ z PP y ^ x Ct z -> x Ct y)

Define (Unitt(x)):   Unitt(x) <-> Continuoust(x) ^ ~ Exist k (k Ct x)

Define: Continuoust(x) <-> ~[Exist y,z (y PP x ^ z PP x ^ y disjoint z ^ ~y Ct z ^ x= y U z]

10.   For all t ~ Exist x (o=Pkgt(x))

11.   For all t,x,y (x=Pkgt(y) ^ Unitt(y) -> atom(x))

Define (Et):  x Et y <-> Exist G (y=Pkgt(G) ^ x P G ^ Unitt(x))

12.   For all t,k,A,B [Pkgt(A) = Pkgk(B) <-> for all z (z Et A <-> z Ek B)]

Define (uPkgt): x= uPkgt(y) <-> Unitt(y) ^ x=Pkgt(y)

Define (uPkg): x= uPkg(y) <-> Exist t (x=uPkgt(y))

Define (unit_package): unit_package(x) <-> Exist y (x=uPkg(y)) or x=o

uPkg is read as "the unit package of".

13.   For all t,x,y [x=Pkgt(y) -> x= U (z| Exist k (k P y ^  z=uPkgt(k)))]

Define (set): set(x) <-> for all y (y O x <-> Exist z ( unit_package(z) ^ y O z))

Define (E): x E y <-> Exist z (z P y ^ z=uPkg(x))

 In English a set is a totality of unit packages, while a member of a set is what is packaged by a unit package that is a part of that set.

Define (element): x is an element <-> Exist y (x E y)

Theorem:  x is an element <-> Exist t (Unitt (x))

14.   For all x (atom(x) -> unit_package(x))

The leads to: all objects are sets, and to: all atoms are singleton sets.

15. For all x E A Exist! y E B Phi(x,y) ^ For all y E B Exist x E A Phi(x,y) -> [A is an element -> B is an element]

16. For all x,y (Phi(x,y) -> x,y are elements) ^ For a,b,c,d (Phi(a,b) ^ Phi(c,d) -> [a=c <-> b=d]) ->   Exist t for all x,y [Phi(x,y) -> Continuoust(x U y)]

/Theory definition finished.

The above system is extensive in some sense, it is illustrative of every bits and pieces about the deep concepts underlying sets and membership. However it can be shortened by letting a lot of the details about temporal contact and temporal units, temporal packages be informal and only add ONE binary relation to that of Part-hood, and that binary relation is "is a unit packaging of". Of course it would be explained at informal level what a unit packaging mean, so if we have 

x is a unit packaging of y

then this would be explained as all constituents of y being connected to each other as to form one continuous whole that is separate from other objects, i.e. y is not a whole of two separate pieces and y is not in external connection with any other object, so that packaging of y is said to be a unit packaging, now if y cannot have its constituents being connected as to form one continuous whole then y or if y fails to be separate from objects disjoint of it, then y is said not to have any unit packaging. Now x itself is the packaging of y so the constituting material of x  itself might not share any part (except bottom) with y. So the packaging of y is the status of distribution of material of y in terms of continuity and discreteness , so when y is a unit then the packaging of y is considered to be a mereological atom, since it is the simplest form of packaging, this not to be confused with the status of y so packaged because y is a unit but not necessarily a mereological atom.

of course the theory would have the same Mereological axioms and definitions (everything in the above exposition down to and including 5).

Now we add the following axioms about packaging.

6. a is a unit packaging of b ^ c is a unit packaging of d <-> [a=c<-> b=d]

7. for all x,y (x is a unit packaging of y -> atom(x))

8. For all x (atom(x) -> Exist y (y is a unit packaging of x))

9. ~ Exist x (o is a unit packaging of x)

Define (unit packaging): x is a unit packaging <-> Exist y (x is a unit packaging of y) or x=o

Define (set): x is a set <-> for all y (atom(y) ^ y P x -> y is a unit packaging)

Define (E): y E x <-> Exist z (z P x ^ z is a unit packaging of y)

Define (element): x is an element <-> Exist y (x E y)

10. For all x,y (Phi(x,y) -> x,y are elements) ^ For a,b,c,d (Phi(a,b) ^ Phi(c,d) -> [a=c -> b=d]) ^ Ua (a| Exist b Phi(a,b)) is an element -> Ub (b|Exist a Phi(a,b)) is an element.

11. Infinity as in Zermelo.

/ Theory definition finished

 We notice here that the last two axioms are the only set axioms, Scheme 10 actually proves: Pairing, Union, Power, Separation and Replacement.  Infinity cannot be derived here as in the above version depending on ideas about continuation of paired elements by a bijective function, this cannot be phrased here, so it needs to be stipulated directly. 

Zuhair 

written at 18/2/2018

The method encourages one to write the following scheme:

Size comprehension scheme: if phi(x,y) is a formula in which only x,y are free, and only free, then

Ax Ez Ay (phi(x,y) -> y=z)

-> 

[ ED(D={x|Exists y phi(x,y)})  

<->

Exists U for all y (Exist x,z (phi(x,z) ^ y in z) -> y in U) ]

is an axiom.

This axiom will prove all axioms of ZF-Regularity-Extensionality-Infinity, ie. it proves all constructive axioms of ZF.