Finitist set theory in ontological modeling

Abstract

This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function as a practical tool in modeling transitive and antitransitive chains of relations without suffering from difficulties of traditional set theories, and a major portion of the functionality of discrete mereology can be incorporated in FST. This makes FST a viable collection theory in ontological modeling.

Keywords

Transitivity intransitivity non-transitivity antitransitivity mereology finitist set theory levels ur-elements aggregates sets computability collection theory layer-cake interpretation vertical-horizontal composition nested structures

1. Introduction

A collection theory is an axiomatic system whose models consist of collections that may be interrelated. When a collection theory is applied in ontological modeling, its formal collections are mapped or assigned to individuals in a domain of application, and collection-theoretical relations are assigned to hold between the individuals. This allows talking about formal-ontological structures of individuals and relations between them. This article develops Finitist Set Theory (FST). FST is a collection theory designed for modeling finite nested structures of individuals and a variety of transitive and antitransitive chains of relations between individuals. The article is organized as follows.

Section 2. The transitivity-antitransitivity-intransitivity distinction is explained. FST is introduced informally by showing how it functions as the logical foundation of the layer-cake interpretation, a general characterization of nested structures found in nature. The layer-cake interpretation brings together many central notions in ontological modeling: vertical-horizontal composition, levels, emergence, types of individuals and relations between individuals. The symbol Γ for vertical composition by nesting-axioms and ∪ for horizontal composition by union-axioms are introduced; Γ-axioms and ∪-axioms are formally defined in Section 3.1 and Section 3.3.

Section 3. The formal structure of FST is introduced. Axioms for complete FST models are given in Section 3.1: extensionality, restriction, singleton sets and union of sets. The concepts subset, proper subset, disjointness, overlap, intersection, union, difference, cardinality, power set, n-member, rank, partition set and transitive closure are defined, and it is proved that weak supplementation is a theorem of FST in Section 3.2. Axioms for some incomplete FST models are given in Section 3.3.

Section 4. FST is applied in modeling nested structures of concrete objects and social constructions.

Section 5. Axioms of discrete mereology (DM) are given in Section 5.1. It is shown how a large portion of the functionality of DM can be incorporated in FST by definitions in Section 5.2. It is shown how DM can be applied as a separate theory on par with FST and how an additional FST model can do the job of DM in Section 5.3.

Section 6. Alternative views about the interconnectedness of transitivity and intransitivity are reviewed in Section 6.1. FST and classical set theories are evaluated as logical foundations of the layer-cake in Section 6.2.

Section 7. The concluding remarks are given.

Appendix: The axioms of complete FST and FST definitions are listed.

2. FST as a logical foundation of the layer-cake

The layer-cake interpretation is a coarse-grained characterization of a wide range of nested structures in nature, which unifies the notions of vertical-horizontal composition, levels, emergence, types and the transitivity-antitransitivity-intransitivity distinction. The layer-cake is not an all-pervasive theory of structures in nature, but it functions as a preliminary test for collection theories that are intended for modeling nested structures.

Transitivity, antitransitivity, intransitivity. The transitivity-antitransitivity-intransitivity distinction is explained in terms of three interrelated distinctions. The first is the dichotomy of transitive and antitransitive chains of one relation between three collections. The chain $x R y R z$ of two successive R relations between three collections x, y, z is called an $R 2$ chain. An $R 2$ chain is either transitive or antitransitive. If both $x R y R z$ and $x R z$ hold simultaneously, the chain is transitive. If $x R y R z$ holds but $x R z$ does not hold, the chain is antitransitive. The second distinction is the trichotomy of transitive, antitransitive and intransitive models.

transitive model:

$\forall x, y, z (x R y R z \to x R z)$ . A model is transitive with respect to $R 2$ chains if $x R y R z \to x R z$ holds for all x, y, z in the model.

antitransitive model:

$\forall x, y, z (x R y R z \to \neg (x R z))$ . A model is antitransitive with respect to $R 2$ chains if $x R y R z$ and $x R z$ do not hold simultaneously for any x, y, z in the model.

intransitive model:

$\exists x, y, z (x R y R z \land x R z) \land \exists x, y, z (x R y R z \land \neg (x R z))$ . A model is intransitive with respect to $R 2$ chains if there exists at least one antitransitive and at least one transitive chain.

The third distinction is the dichotomy of collection theories with transitive and intransitive basic relations. A theory whose basic relation is transitive is a transitive theory; Boolean algebras and traditional mereologies are transitive theories (Section 5.1). A theory whose basic relation is intransitive is an intransitive theory; FST and traditional set theories are intransitive theories (Section 6.2). By a transitive basic relation such as the mereological part relation (⪯), only transitive chains of relations can be modeled, such as $a ⪯ a b ⪯ a b c$ . By an intransitive basic relation such as the set theoretic membership relation (∈), one can model transitive 2-chains of membership such as $a \in {a} \in {a, {a}}$ , as well as antitransitive 2-chains such as $a \in {a} \in {{a}}$ . One can also apply membership in defining other relations such as subset and overlap (Section 3.2) and a relation similar to the mereological part relation (Section 5.2). It is an essentially unifying feature of FST that interdependencies of all relations that are defined in terms of membership are implicit in the definitions.

Vertical-horizontal composition and levels. In vertical composition the level rises along the ascension in the hierarchy, where two or more lower-level individuals are fused into a whole whose level is one degree higher than the highest level of the fused lower-level individuals. For instance, two individuals of level n are vertically composed into a level $n + 1$ individual; one individual of level n and one of level $n - 1$ are likewise vertically composed into a level $n + 1$ individual. In horizontal composition individuals are fused together in such a way that the level of the resulting whole is the same as the level of the highest-level fused individual: e.g. two individuals of levels n and $n - 1$ are horizontally composed into a level n individual.

The overall scheme of vertical-horizontal composition applies to concrete objects and social constructions, which are the two overall types of individuals modeled in Section 4. Chains of vertical relations between lower-level and higher-level concrete objects are typically or always antitransitive; chains of vertical relations between lower-level and higher-level social constructions may be antitransitive or transitive, or combinations of antitransitive and transitive 2-chains. Horizontal chains of relations are always transitive. Fine (2010, p. 578) uses the terms ‘horizontal’ and ‘vertical’ and concludes that he is “inclined to believe that this method for deriving forms of composition is of general application.” According to Wimsatt, the constituents of a composite have typically the same level or their levels are close to one another:

…compositional levels of organization are the simplest general and large scale structures for the organization of matter. They are constituted by families of entities usually of comparable size and dynamical properties, which characteristically interact primarily with one another, and which, taken together, give an apparent rough closure over a range of phenomena and regularities. (Wimsatt, 2006, p. 222)

In FST, the level of an individual finds a formal foundation from the nesting level or rank of the set by which the individual is modeled (Section 3.2). If the putative level of basic elements or ur-elements a, b, c, d is 0, the level of sets ${a, b}$ and ${b, c}$ is 1 as the ur-elements are nested by a single set; the level of set ${{a, b}, {c, d}}$ is 2 as the ur-elements are nested by two concentric sets, and so forth. The level n of individual x is one step higher than the level of the highest-level member of x. Therefore, at least one member of x is on level $n - 1$ ; the other member(s) may in principle be on any level that is lower than n. This is congenial with Bunge (1979, ch. 1.5) who states that a thing belongs to level $n ⩾ 1$ if and only if it is composed of things in some or all of the preceding levels. However, in examples in Section 4 the members of a level n individual are typically on the same level $n - 1$ or their levels are close to one another.

Vertical chains or links between levels are modeled by the membership relation, denoted by the symbol ∈ (epsilon), where $a \in {a, b}$ is read as ‘a is a member of ${a, b}$ ,’ and $c \notin {a, b}$ is read as ‘c is not a member of ${a, b}$ .’ A chain of two successive vertical links is either transitive such as $a \in {a} \in {a, {a}}$ , or antitransitive such as $a \in {a} \in {{a}}$ . Horizontal links are modeled by subset relations, denoted by the symbol ⊆, and ⊂ for proper subset (Section 3.2). A chain of horizontal links is always transitive, such as $a \subset {a, b} \subset {a, b, c}$ .

Composition principles. Composition principles can be seen as functions which take scientific data about the target domain as input, and which output a formal model of collections which corresponds to individuals in the target domain. FST’s set construction axioms are formal analogs of the composition principles. When consistent data is given as input to the selected axioms, a FST model follows as output. Consider composition principles for concrete objects. The symbol Γ for vertical composition and the symbol ∪ for horizontal composition are obtained from Fine (2010). The Γ-axioms and the ∪-axioms are formally defined in Section 3.3.

Γ Vertical composition by nesting-axioms. Vertical composition takes place when the interrelated constituents of a composite object are of a different type or of different types than the composite; in this case the composite is said to be emergent and on a higher level than its constituents.1

¹
The view that a higher-level object consists of interrelated lower-level parts does not have to be coupled with reductionism: “this is not a reductionistic analysis in the sense in which that term might be used by a philosopher” (Wimsatt, 2006, p. 223).

This is analogous to Kim’s (1999, p. 20) characterization of emergence of complex higher-level entities: “Systems with a higher-level of complexity emerge from the coming together of lower-level entities in new structural configurations.” Bunge (1960, p. 399) gives an analogous characterization: “The lower order wholes are the building blocks of the higher order ones: the latter emerge through the harmonious action (interaction) of lower order individual units.” Applying FST, if constituents y and z of a type D object x are not type D objects nor of subtypes of D, and if the relation between y and z is not of the same type as the relations between the constituents of y and not of the same type as the relations between the constituents of z, then x is modeled as a set and its constituents y and z as members of x, where

x = {y, z}

. FST’s axioms for vertical composition are called nesting-axioms, abbreviated as Γ-axioms. The overall form of the nesting-axioms is

Γ (r_{1}, \dots, r_{n}) : = {r_{1}, \dots, r_{n}}

, which means that the axiom produces a set that nests its 1 to n inputs (urs or sets or both). For instance:

Γ (a) : = {a}

;

Γ ({a}) : = {{a}}

;

Γ ({{{a}}}) : = {{a}}

;

Γ (a, b) : = {a, b}

;

Γ ({a, b}, {c, d}) : = {a, b, c, d}

∪ Horizontal composition by union-axioms. Horizontal composition is non-emergent. The composite object is of the same type as its constituents, and is not on a higher level than its highest-level constituent. Applying FST, when the constituents y and z of a type D object x are of the same type D, then x is modeled as a set whose constituents are proper subsets of x. For instance, when $y = {a, b}$ and $z = {c, d}$ are the constituents of $x = {a, b, c, d}$ , the proper subset relations hold: ${a, b} \subset {a, b, c, d}$ ; ${c, d} \subset {a, b, c, d}$ . FST’s axioms for horizontal composition are called union-axioms, abbreviated as ∪-axioms.

Vertical-horizontal composition. We can talk about constituents of an individual in two compatible ways: looking horizontally, the constituents of ${a, b, c, d}$ are its subsets ${a, b}$ and ${c, d}$ ; looking vertically, the constituents of ${a, b, c, d}$ are its members a, b, c, d. As the membership relations are generic, all composition can be seen as vertical, but as in many cases vertical composition can be seen also from a horizontal angle, it is intelligible to call composition generally as vertical-horizontal. The construction of ${a, b, c, d}$ by the nesting-axiom $Γ (a, b, c, d) : = {a, b, c, d}$ is vertical in the sense that a, b, c, d are members of ${a, b, c, d}$ , but horizontal in the sense that we can also apply the union-axiom $\cup ({a, b}, {c, d}) : = {a, b, c, d}$ . Even this is vertical-horizontal, as ${a, b}$ and ${c, d}$ must either be composed vertically, or horizontally from ${a}$ , ${b}$ , ${c}$ , ${d}$ which must be composed vertically.

When data is given as input to the composition principles, type hierarchies follow as outputs: (i) hierarchies of determinable types; (ii) hierarchies of determinate types; (iii) supertype-subtype hierarchies. Hierarchies of individuals are exemplified in Section 4.

The definition ‘molecules consist of two or more atoms interrelated by chemical bonds’ defines the determinable type ‘molecule’. The definition ‘intermolecular composites consist of two or more molecules interrelated by intermolecular bonds’ defines the determinable type ‘intermolecular composite’. Thereby, we have a 2-level hierarchy of determinable types, where intermolecular composite is on level 2, molecule on level 1, and atom on level 0.2

0 is putative and deduced from the given type definitions. The assignment of a level to a concrete object is always at most putative, for the full structure and thus the genuine level of an object cannot be known in principle. It can always be debated whether there really exists the deepest level 0, or whether an object is infinitely divisible, but this debate does not touch practical modeling initiatives where we must in any case assign a putative level 0. Even if nature were spatially infinite or infinitely divisible or both, this could not be known even in principle, and even if this were supposed to be true, still only finitely many parts and levels could actually be known and computed. Masolo agrees: “From an applicative perspective, I don’t consider the existence of bottom levels as very restrictive (infinite (down) chains of objects are impractical)” (Masolo, 2010, p. 260).

The definition ‘a water molecule consists of two hydrogen atoms and one oxygen atom, where each hydrogen is bonded to the oxygen by a single covalent bond’ defines the absolutely determinate type ‘water molecule’. Thereby, we have a 1-level hierarchy, a determinate type, where the water molecule is on level 1 and its constituent atoms are on level 0.

A subtype is more determinate than its supertype. Subtype-supertype hierarchies can be modeled e.g. by FST’s proper subset relation such as in noble gas ⊂ atom, where noble gas is determinate relative to atom, but noble gas is not absolutely determinate, for it is composed of several absolutely determinate types such as helium and neon (cf. Armstrong 1978, II, pp. 112, 117).

3. Axioms and definitions of FST

FST models are of type ${U_{α}, S_{β}, \in}$ , which is abbreviated as $M_{α, β}$ . $U_{α}$ is the collection of ur-elements of model $M_{α, β}$ . Ur-elements (urs) are indivisible primitives. By assigning a finite integer such as 2 as the value of α, it is determined that $U_{α}$ contains exactly 2 urs. $S_{β}$ is a collection whose elements will be called sets. $β ⩾ 0$ is a finite integer which denotes the maximum rank (nesting level) of sets in $S_{β}$ . Every set in $S_{β}$ has one or more sets or urs or both as members. The assigned α and β and the applied axioms fix the contents of $U_{α}$ and $S_{β}$ . To facilitate the use of language, expressions such as “sets that are elements of $S_{β}$ of model $M_{α, β}$ and urs that are elements of $U_{α}$ of model $M_{α, β}$ ” are abbreviated as “sets and urs that are elements of $M_{α, β}$ ”.

FST’s formal development conforms to its intended function as a tool in ontological modeling. The goal of an engineer who applies FST is to select axioms which yield a model that is one-one correlated with a target domain that is to be modeled by FST, such as a range of chemical compounds or social constructions that are found in nature (Section 4). The target domain gives the engineer an intuition about the contents of the FST model that ought to be one-one correlated with it. FST provides a framework that facilitates selecting specific axioms that yield the one-one correlation. The axioms of extensionality and restriction (Section 3.1) are postulated in all versions of FST, but set construction axioms (nesting-axioms and union-axioms) vary; the assignment of finite integer values to α and β is implicit in the selected set construction axioms.

FST is thereby not a single theory, but a name for a family of theories or versions of FST, where each version has its own set construction axioms and a unique model $M_{α, β}$ , which has a finite cardinality and all its sets have a finite rank and cardinality. FST axioms are formulated by first-order logic complemented by the member of relation ∈. All versions of FST are first-order theories.3

³
For, the axioms of a particular FST model apply only a finite number of formulas with a finite lenght. In contrast, e.g. the theory of all complete FST models ${{M_{α, β}, \in} ∣ α, β \in N, α ⩾ 1, β ⩾ 0}$ , where the axiom of singleton sets is an axiom schema, would be a second-order theory.

In the axioms and definitions, symbols x, y, z, v, w are variables for sets, r, s, t are variables for both sets and urs, u is a varible for urs, and a, b, c, d denote individual urs of a model. The symbols for urs may appear only on the left side of ∈. The symbols for sets may appear on both sides of ∈. Examples of valid formulas that have a truth-value:

x \in y

r \in y

u \in y

An applied FST model is always the minimal model which satisfies the applied axioms. This guarantees that those and only those elements exist in the applied model which are explicitly constructed by the selected axioms: only those urs exist which are stated to exist by assigning their number, and only those sets exist which are constructed by the selected axioms; no other elements exist in addition to these. This interpretation is needed, for typical FST axioms which generate e.g. exactly one set ${a}$ do not otherwise exclude sets such as ${{a}}, {{{a}}}, \dots$ .

Cumulative construction axioms for complete FST models are introduced in Section 3.1. A cumulative axiom (or several axioms which are together cumulative) may take its own outputs as inputs. For instance, the axiom of singleton sets is cumulative: given ur a and $β > 0$ , the axiom of singleton sets implies the existence of ${a}$ ; given ${a}$ and $β > 1$ , the axiom implies the existence of ${{a}}$ ; and so on up to the assigned β. Cumulative and non-cumulative construction axioms for incomplete FST models are introduced in Section 3.3 and applied in Section 4.

3.1. Axioms for complete FST models

Complete FST models contain all permutations of sets and urs within the limits of α and β. The axioms for complete FST models are extensionality, restriction, singleton sets and union of sets. Extensionality and restriction are axioms of all versions of FST, whereas the axiom for singleton sets is a provisional nesting-axiom (Γ-axiom) and the axiom of union of sets is a provisional union-axiom (∪-axiom).

ax. extensionality:

$\forall r (r \in x \leftrightarrow r \in y) \leftrightarrow x = y$ . Set x is identical to set y iff (if and only if) x and y have the identical members, may these be sets, urs or both.

ax. restriction:

$\forall x \exists r (r \in x)$ . Every set has either a set or an ur as a member. The empty set {} has no members, and therefore there exists no such thing as {} in FST. Urs are the only ∈-minimal elements in FST. Every FST set contains at least one ur as the ∈-minimal member on the bottom.

ax. singleton sets:

$\forall r_{< β} \exists x \forall s (s = r \leftrightarrow s \in x)$ . For every ur and set r that has a rank smaller than β, there exists the singleton set $x = {r}$ . The rank restriction ( $r_{< β}$ , Section 3.2) in the axiom does the job of the axiom of foundation of traditional set theories: constraining the rank of sets to an assigned finite β entails that there are no non-wellfounded sets, for such sets would have a transfinite rank. Given urs a and b in $M_{2, 1}$ , the axiom of singleton sets generates only sets ${a}$ and ${b}$ , whereas the axiom of pairing of traditional set theories generates ${a}$ , ${b}$ and ${a, b}$ .

ax. union of sets:

$\forall x \forall y \exists z \forall r ((r \in x \lor r \in y) \leftrightarrow r \in z)$ . For all sets x and y, there exists set z which contains as members all those and only those sets and urs that are members of x, members of y, or members of both x and y. For instance, if sets ${a}$ and ${b}$ exist, the axiom of union of sets states that the set ${a, b}$ exists. If sets ${a, b}$ and ${b, c}$ exist, the axiom states that ${a, b, c}$ exists. If ${a}$ and ${{b}}$ exist, the axiom states that ${a, {b}}$ exists. The difference to the axiom of union of traditional set theories is explained in Section 6.2.

Constructing sets. Consider three examples of building sets by applying the axioms of singleton sets which is abbreviated as $Γ (r)$ and union of sets which is abbreviated as $\cup (x, y)$ . That $Γ (r)$ produces set ${r}$ is written as $Γ (r) : = {r}$ . Building a set by FST axioms is a proof of its existence in a FST model.

Let us construct set ${a}$ which exists in $M_{1, 1}$ . Assigning $α = 1$ gives ur a. $Γ (a) : = {a}$ .

Let us construct set ${a, b, c}$ which exists in $M_{3, 1}$ . Assigning $α = 3$ gives urs a, b, c. $Γ (a) : = {a}$ , $Γ (b) : = {b}$ , and $Γ (c) : = {c}$ . $\cup ({a}, {b}) : = {a, b}$ , and $\cup ({a, b}, {c}) : = {a, b, c}$ .

Let us construct set ${a, {{b}}}$ which exists in $M_{2, 3}$ . Assigning $α = 2$ gives urs a, b. $Γ (a) : = {a}$ , $Γ (b) : = {b}$ , $Γ ({b}) : = {{b}}$ , and $Γ ({{b}}) : = {{{b}}}$ . $\cup ({a}, {{{b}}}) : = {a, {{b}}}$ .

To illustrate that all permutations of sets are included in complete FST models, consider model $M_{n, 1}$ with 1 as the maximum rank and an arbitrary number of urs $a_{1}, a_{2}, \dots, a_{n}$ . The axiom of singleton sets gives singletons of every ur: ${a_{1}}, {a_{2}}, \dots, {a_{n}}$ , i.e., we have all level 1 sets of cardinality 1. The axiom of union gives a union of every two existing sets, and cumulatively operates also with its own outputs. The generation of sets can be characterized in stages: the generation of unit sets is stage I; in stage II, all level 1 sets of cardinality 2 are generated by unions of every two level 1 unit sets; in stage III, all level 1 sets of cardinalities 3 and 4 are generated by unions of the products of stages I–II; in stage IV, all level 1 sets of cardinalities 5, 6, 7, 8 are generated by unions of the products of stages I–III. This characterization qualifies as an informal consistency proof of FST. Consider examples of existing sets and urs in $M_{α, β}$ with some assigned α and β.

One ur a exists.

Two urs a, b exist.

One ur a and the set ${a}$ exist.

Two urs a, b and sets ${a}$ , ${b}$ , ${a, b}$ exist.

One ur a and sets ${a}$ , ${{a}}$ , ${a, {a}}$ exist.

The recursive formula $sets (α, β)$ gives the number of sets in $M_{α, β}$ :

$sets (α, 0) = 0$ .

$sets (α, 1) = 2^{α} - 1$ .

$sets (α, β) = 2^{α + sets (α, β - 1)} - 1$ .

In $M_{2, 2}$ there are $sets (2, 2) = 2^{2 + sets (2, 1)} - 1 = 2^{2 + 3} - 1 = 31$ sets.

In $M_{2, 3}$ there are $sets (2, 3) = 2^{2 + 31} - 1 = 2^{33} - 1$ sets.

As urs are counted in, the cardinality of $M_{α, β}$ is $sets (α, β) + α$ .

3.2. FST definitions

FST definitions should be understood as practical naming conventions which are used in stating that the elements of an applied FST model are or are not interrelated in specific ways. The definitions ought not be seen as axioms: only axioms entail existence of elements of a FST model, not definitions. In order to avoid conflicts, the definitions must be subjugated to the applied axioms with the given α and β. To illustrate a seeming conflict, suppose that ${a, b}$ and ${b, c}$ are the only sets of the applied model. The definition of intersection states that ${a, b} \cap {b, c} = {b}$ . As ${b}$ does not exist in the applied model, the definition appears to be an axiom. This is only apparent, for ${b}$ does not have to exist in order to state that the only common element of ${a, b}$ and ${b, c}$ is b. Similarly with all definitions. def. subset:

$\forall r (r \in x \to r \in y)$ , is denoted as $x \subseteq y$ . x is a subset of y iff every member of x is a member of y. Examples: ${a} \subseteq {a}$ ; ${a, b} \subseteq {a, b, c}$ . That x is not a subset of y is written as $x ⊈ y$ . Examples: ${a} ⊈ {b}$ ; ${a, b} ⊈ {b, c}$ . Due to the exclusion of the empty set ${}$ , in FST $x \subseteq y$ means that all members of x are members of y, and there exists at least one member in x and at least one member in y. In traditional set theories where ${}$ exist, $x \subseteq y$ means that x does not have any members that are not members of y; either all members of x are members of y, or x does not have any members, i.e., ${} \subseteq y$ holds for every y.

def. proper subset:

$x \subseteq y \land y ⊈ x$ , is denoted as $x \subset y$ . x is a proper subset of y iff x is a subset of y and y is not a subset of x. Examples: ${a} \subset {a, b}$ ; ${a, b} \subset {a, b, c}$ . That x is not a proper subset of y is written as $x ⊄ y$ . Examples: ${a} ⊄ {a}$ ; ${a, b, c} ⊄ {a, b}$ . In FST, $x \subset y$ means that all members of x are members of y, there exists at least one member in x, at least two members in y, and at least one member of y is not a member of x. In traditional set theories, $x \subset y$ means that x does not have any members that are not members of y, and y has at least one member that is not a member of x, i.e., ${} \subset y$ holds for every $y \neq {}$ .

def. disjointness:

$∄ r (r \in x \land r \in y)$ , is denoted as $x ≀ y$ . x and y are disjoint iff they do not have any common members. Examples: ${a} ≀ {b}$ ; ${a} ≀ {{a}}$ .

theorem of weak supplementation:

$x \subset y \to \exists z (z \subset y \land z ≀ x)$ . (cf. Varzi 2016). Weak supplementation (WS) expresses that a proper subset x of y is not the whole y, but must be supplemented by z to compose y, where z and x are disjoint. In FST, when $x \subset y$ holds, y has another subset z that is disjoint with x. For instance, ${a} \subset {a, b} \to ({b} \subset {a, b} \land {b} ≀ {a})$ is true in all FST models which contain the set ${a, b}$ . Varzi maintains that it “seems appropriate to regard” WS “as providing a minimal but substantive addition to” core mereology (the fusion of the axioms of reflexivity, transitivity and antisymmetricity). As a theorem, WS is nothing additional to FST.

def. overlap:

$\exists r (r \in x \land r \in y)$ , is denoted as $x \circ y$ . x and y overlap iff they have one or more common members. Examples: ${a} \circ {a}$ ; ${a, b} \circ {b, c}$ . Disjointness is the contrary of overlap: $x \circ y \leftrightarrow \neg (x ≀ y)$ ; $\neg (x \circ y) \leftrightarrow x ≀ y$ .

def. intersection:

$\forall r ((r \in x \land r \in y) \leftrightarrow r \in z)$ , is denoted as $z = x \cap y$ . The intersection of x and y, $z = x \cap y$ , contains those and only those sets and urs that are members of both x and y. Examples: ${a} \cap {a} = {a}$ ; ${a, b, c} \cap {b, c, d} = {b, c}$ . As the empty set does not exist in FST, the intersection of two disjoint sets does not exist. When $r \neq s$ , ${r} \cap {s} = y$ is not true for any y. In this case, the disjointness relation ≀ can be used: ${r} ≀ {s}$ . In traditional set theories the intersection of two disjoint sets is the empty set: ${a} \cap {b} = {}$ . Were the axiom of restriction deleted from FST and were the empty set postulated, this would not imply that the empty set is the intersection of two disjoint sets.

def. union:

$\forall r ((r \in x \lor r \in y) \leftrightarrow r \in z)$ , is denoted as $z = x \cup y$ . Set z contains as members all those sets and urs that are members of x, members of y, or members of both x and y. Examples: ${a} \cup {a} = {a}$ ; ${a, b} \cup {b, c} = {a, b, c}$ ; ${a} \cup {{b}} = {a, {b}}$ . The use of ∪ as a relation symbol must be distinguished from its use in union-axioms.

def. difference:

$\forall r (r \in z \leftrightarrow (r \in x \land r \notin y))$ , is denoted as $z = x ∖ y$ . The difference z of x and y contains every member of x that is not a member of y. Examples: ${a} ∖ {b} = {a}$ ; ${a, b, c} ∖ {a, b} = {c}$ . As the empty set does not exist, it cannot be stated that ${x} ∖ {x} = {}$ . If x is a subset of y, there does not exist z such that $z = x ∖ y$ : $\forall x, y (x \subseteq y \leftrightarrow ∄ z (z = x ∖ y))$ .

def. cardinality.

Cardinality denotes the number of members of a set. Cardinality is defined only for sets: urs do not have a cardinality. The cardinality of ${r}$ is 1, disregarding whether r is a set or an ur-element. The lowest possible cardinality of an FST set is 1, whereas in traditional set theories the cardinality of {} is 0. $card (x) = n$ means that the cardinality of set x is n. E.g. $card ({y, z}) = 2$ , $card ({x, y, {z}}) = 3$ , $card ({x, {x}, {{x}}}) = 3$ , and $card ({x, y, z, w}) = 4$ .

$card (x) = 1$ is defined as: $\exists s (s \in x \land \forall r (r \in x \leftrightarrow r = s))$ .

$card (x) ⩾ n$ , where $n ⩾ 2$ , is def. as: $\exists s_{1}, s_{2}, \dots, s_{n} ((⋀_{k = 1}^{n} s_{k} \in x) \land ⋀_{a = 1}^{n - 1} ⋀_{b = a + 1}^{n} s_{a} \neq s_{b})$ .

$card (x) = n$ , where $n ⩾ 2$ , is defined as: $card (x) ⩾ n \land \neg (card (x) ⩾ n + 1)$ .

def. power set:

$\forall z (z \in y \leftrightarrow z \subseteq x)$ , denoted as $y = P (x)$ . Examples: $P ({r}) = {{r}}$ ; $P ({r, s}) = {{r}, {s}, {r, s}}$ . Power sets in FST do not contain the empty set, and thus $card (P (x)) = 2^{card (x)} - 1$ . In FST power set is not required in building sets, whereas e.g. in ZF set theory the axiom of power set is essential in building the hierarchy transfinite sets. In ZF power sets contain the empty set, e.g. as in $P ({y, x}) = {{}, {y}, {z}, {y, z}}$ , which makes $card (P (x)) = 2^{card (x)}$ .

def.n-member:

$r \in_{1} x$ is defined as $r \in x$ .

$r \in_{2} x$ is defined as $\exists y (r \in y \in x)$ .

$r \in_{n} x$ , where $n ⩾ 2$ , is defined as $\exists y (r \in_{n - 1} y \in x)$ .

That

r \in_{1} x

holds can be stated by saying that r exists in the first partition level of x. That

r \in_{2} x

holds can be stated by saying that r exists in the second partition level of x. And so forth. The term ‘partition level’ and the recursive definition of n-member are adapted from (Seibt, 2015, pp. 178–80) and (Seibt, 2009, Section 3.2).

def. rank.

The rank of a set is the formal analog of the level of an individual. That the rank of set x is n, is written as $rank (x) = n$ , and abbreviated as $x_{β}$ in some nesting-axioms. As a convention, the rank of an ur-element is 0. As there is no empty set in FST, the smallest possible rank of a FST set is 1, whereas in traditional set theories the rank of {} is 0. The rank of set z is defined as the greatest nesting level of all ∈-minimal elements of z. The rank of ${a}$ is 1, as the nesting level of a in ${a}$ is 1. The rank of ${{a}}$ is 2, as a is nested by two concentric sets. The rank of ${{a}, a}$ is 2, as 2 is the greatest nesting level of all ∈-minimal elements of ${{a}, a}$ . The rank of ${{{a}}}$ is 3, the rank of ${{{{a}}}}$ is 4, and so on. Rank can be defined by applying the definition of n-member:

$rank (s) = 0$ is defined as: s is an ur-element.

$rank (r) = n$ is defined as: $\exists u (u \in_{n} x) \land ∄ u (u \in_{n + 1} x)$ .

def.

[n m]

members.

$r \in_{[n m]} x$ , where $n ⩽ m$ , is defined as: $r \in_{n} x \lor r \in_{n + 1} x \lor r \in_{n + 2} x \lor \dots \lor r \in_{m} x$ .

r is an n-to-m member of x when r is an n-member of x or an $n + 1$ -member of x or …or an m-member of x.

def. partition set.

A partition set that contains all n-members of a set is defined as:

${partition}_{1} (x) = x$ .

${partition}_{2} (x) = y$ is defined as: $\forall r (r \in_{2} x \leftrightarrow r \in y)$ .

${partition}_{n} (x) = y$ is defined as: $\forall r (r \in_{n} x \leftrightarrow r \in y)$ .

def. transitive closure:

$\forall r (r \in_{[1 rank (x)]} x \leftrightarrow r \in y)$ , denoted as $y = T c (x)$ . $y = T c (x)$ means that set y is the transitive closure of set x. y contains all sets and urs of the input set x, i.e., the whole inner structure of x. Examples: $T c ({{a, b}}) = {a, b, {a, b}}$ ; $T c ({{{a}}}) = {a, {a}, {{a}}}$ ; $T c ({a, {a}}) = {a, {a}}$ .

3.3. Axioms for incomplete FST models

One or more sets that exist in a complete model $M_{α, β}$ do not exist in an incomplete model $M_{α, β}$ , with the same α and β. While complete models are built by the axiom of singleton sets (a Γ-axiom) and the axiom of union of sets (a ∪-axiom), incomplete models are built by Γ-axioms and ∪-axioms which are complemented by restrictions that exclude sets that are not needed in an applied model. Cumulative and non-cumulative construction axioms for incomplete FST models are exemplified. The non-cumulative constructions axioms that are exemplified here function as logical forms of some axioms that are applied in Section 4. That an axiom entails the existence of set x is abbreviated as $AXIOM : = x$ .

Union-axioms. The axiom of union of sets is first modified into R-union of two sets which gives unions of two sets which stand in relation R. R-union of two sets is extended into R-union of n sets, and examples of particular R-unions are given.

ax. r-union of two sets:

$\forall x, y (R (x, y) \leftrightarrow (\exists z \forall r ((r \in x \lor r \in y) \leftrightarrow r \in z)))$ . For all sets x and y which stand in relation R, there exists set z which contains as members all those and only those sets and urs that are members of x, members of y, or members of both x and y. That z is the R-union of x and y is abbreviated as $\cup (x, y ∣ R (x, y)) : = z$ .

ax. r-union of n sets:

$\forall x_{1}, \dots, x_{n} (R (x_{1}, \dots, x_{n}) \leftrightarrow (\exists z \forall r ((r \in x_{1} \lor \dots \lor r \in x_{n}) \leftrightarrow r \in z)))$ . For all sets $x_{1}, \dots, x_{n}$ which stand in relation R, there exists set z which contains as members all those and only those sets and urs that are members of one or more of $x_{1}, \dots, x_{n}$ . That z is the R-union of sets $x_{1}, \dots, x_{n}$ is abbreviated as $\cup (x_{1}, \dots, x_{n} ∣ R (x_{1}, \dots, x_{n})) : = z$ .

ax. union of two sets with identical rank:

$\cup (x, y ∣ rank (x) = rank (y)) : = z$ . Set z contains as members only those sets and urs that are members of x or y or both, and where the rank of x is identical to the rank of y. In model $M_{2, 3}$ whose only construction axioms are union of sets with identical rank and singleton sets, sets such as ${a, b}$ and ${{a}, {b}}$ exist, but e.g. ${{a}, {{b}}}$ and ${{b}, {{a}}}$ do not because their members have different ranks. All membership chains produced by this combination of axioms are antitransitive, with all α and β. The recursive formula for the number of sets in $M_{α, β}$ becomes:

$sets (α, 1) = 2^{α} - 1 .$

$sets (α, β) = sets (α, β - 1) + 2^{sets (α, β - 1)} - 1$ .

For instance, the number of sets in

M_{2, 2}

sets (2, 2) = sets (2, 1) + 2^{sets (2, 1)} - 1 = 4 - 1 + 8 - 1 = 10

. The sets in

M_{2, 2}

are

{a}

;

{b}

;

{a, b}

;

{{a}}

;

{{b}}

;

{{a, b}}

;

{{a}, {b}}

;

{{a, b}, {a}}

;

{{a, b}, {b}}

;

{{a, b}, {a}, {b}}

ax. union of two sets with no identical urs:

$\cup (x, y ∣ ∄ u (u \in_{[1 rank (x)]} x \land u \in_{[1 rank (y)]} y)) : = z$ . Set z contains as members only those sets and urs that are members of x or y or both, and where x and y contain no identical urs in any level. In model $M_{2, 2}$ whose only construction axioms are singleton sets and union of two sets with no identical urs, all sets which contain the identical ur twice in some level of the hierarchy are discluded. Sets such as ${a, b}$ and ${{a}, {b}}$ and ${a, {b}}$ exist in $M_{2, 2}$ , but sets such as ${{a, b}, {b}}$ and ${a, {a}}$ do not exist because they have members which contain identical urs, or one of their members is an ur that exists in some level of the other member. All membership chains produced by this combination of axioms are antitransitive, with all α and β. Allowing sets which contain two or more identical urs allows transitive membership chains such as $a \in {a} \in {a, {a}}$ which may be needed in modeling social constructions (Section 4).

ax. union of two sets with identical rank and no identical urs:

$\cup (x, y ∣ rank (x) = rank (y) \land ∄ u (u \in_{[1 rank (x)]} x \land u \in_{[1 rank (y)]} y)) : = z$ . Model $M_{2, 2}$ whose only construction axioms are singleton sets and union of two sets with identical rank and no identical urs, contains only 7 sets: ${a}$ ; ${b}$ ; ${a, b}$ ; ${{a}}$ ; ${{b}}$ ; ${{a, b}}$ ; ${{a}, {b}}$ .

Nesting-axioms. Γ-axioms can be formulated with one to n inputs. $Γ (r) : = {r}$ is the axiom of singleton sets, which generates singleton sets of all sets and urs in the applied model whose rank is below β. $Γ (r, s) : = {r, s}$ produces pair sets of all two unidentical sets or urs in the applied model whose rank is below β. Similarly as the ∪-axioms, the Γ-axioms can be complemented by adding restrictions R for the types of the nested elements and relations that need to hold between them. Consider the logical form of R-pairing of two unidenticals, $Γ (r, s ∣ R) : = {r, s}$ :

ax. ax. r-pairing of two unidenticals:

$\forall r_{< β}, s_{< β} (r \neq s \land R \to \exists x \forall t ((t = r \lor t = s) \leftrightarrow t \in x))$ . For all two unidentical sets or urs r and s in the given model, whose rank is below β and who satisfy constraints R, there exists set $x = {r, s}$ whose only elements are r and s. For instance, $Γ (r, s ∣ J (r) \land J (s) \land S (r, s)) : = {r, s}$ states: for all two unidentical sets or urs r and s which are of the type J and interrelated by S, there exists set $x = {r, s}$ . As another example, the axiom of pairing of unidenticals with equal rank $Γ (r, s ∣ rank (r) = rank (s)) : = {r, s}$ states: for every two unidentical urs or sets r and s that have the same rank smaller than β, there exists the pair set $x = {r, s}$ . In a model whose only construction axioms are pairing of unidenticals with equal rank and union of sets, the minimum cardinality of a set is 2. The axiom of pairing is first extended into the axiom of nesting n unidenticals, and then complemented by constraints R.

ax. nesting of n unidenticals:

$Γ (r_{1}, \dots, r_{n}) : = {r_{1}, \dots, r_{n}}$ states that there exists all sets that contain exactly n sets or urs $r_{1}, \dots, r_{n}$ in the applied model which have rank smaller than β: $\begin{matrix} \forall r_{1_{< β}}, \dots, r_{n_{< β}} [(\underset{i \neq j}{⋀} r_{i} \neq r_{j}) \to \exists x \forall t ((⋁_{i = 1}^{n} t = r_{i}) \leftrightarrow (t \in x))] . \end{matrix}$

ax. r-nesting of n unidenticals:

$Γ (r_{1}, \dots, r_{n} ∣ R) : = {r_{1}, \dots, r_{n}}$ states that there exists all sets that contain exactly n sets or urs $r_{1}, \dots, r_{n}$ in the applied model which have rank smaller than β, and which have specific types and are interrelated in specific ways, as stated by R: $\begin{matrix} \forall r_{1_{< β}}, \dots, r_{n_{< β}} [((\underset{i \neq j}{⋀} r_{i} \neq r_{j}) \land R) \to \exists x \forall t ((⋁_{i = 1}^{n} t = r_{i}) \leftrightarrow (t \in x))] . \end{matrix}$

Combinatorial implosion. In traditional set theories and complete FST, the number of rank n sets increases exponentially as n approaches the maximum rank β. In contrast, in some incomplete FST models the number of rank $β - 1$ sets is greater than the number of rank β sets, at least with some small α. Consider the following construction axioms: pairing of unidenticals with equal rank; union of sets with identical rank and without identical urs. By these axioms, $M_{4, 2}$ contains altogether 11 level 1 sets, which are permutations of urs a, b, c, d : ${a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d}; {a, b, c}; {a, b, d}; {a, c, d}; {b, c, d}; {a, b, c, d}$ . $M_{4, 2}$ contains only three level 2 sets: ${{a, b}, {c, d}}$ ; ${{a, c}, {b, d}}$ ; ${{a, d}, {b, c}}$ . One way to characterize the combinatorial implosion is that the existence and formation of level β sets requires the existence of at least $α = 2^{β}$ urs. Min 2 urs are needed in forming a level 1 set, min 4 urs for forming level 2 sets, min 8 urs for level 3 sets, and so forth. This means that e.g. $M_{2, 1}$ is identical to $M_{2, 2}, M_{2, 3}, M_{2, 4}$ and $M_{2, β}$ with all $β ⩾ 1$ : the only set in these models is ${a, b}$ even when the assigned maximum rank is higher than 1. Likewise, $M_{4, 2}$ is identical to $M_{4, β}$ with all $β ⩾ 2$ : the highest-level sets of all these models are the rank 2 sets ${{a, b}, {c, d}}$ ; ${{a, c}, {b, d}}$ ; ${{a, d}, {b, c}}$ , even when the assigned maximum rank is higher than 2.

4. Applying FST in ontological modeling

FST is applied in ontological modeling by mapping collections of an FST model to individuals (concrete objects and social constructions) in a target domain, and by assigning collection-theoretical relations between the individuals. That a collection is mapped or assigned to an individual means that a collection is equated, similarized or analogized with a certain aspect of the structure of the individual, and that the individual is described, modeled or characterized by the collection. Once a collection has been mapped to an individual, we can talk about the formal-ontological properties of the individual, such as that the individual has certain members or parts. Consider two approaches to modeling these by FST: (1) propagation of a model by cumulative axioms; (2) by one or more axioms per one individual.

In (1), a FST model is generated by cumulative axioms, after which individuals in a target domain are mapped to sets and urs in the model. (1) can be seen as propagation towards perfect generative axioms, where the ideal goal is to generate a model whose sets are one-one correlated with individuals of a target domain. For instance, the goal may be to formulate axioms which generate a model whose sets are one-one correlated with all physically possible molecules. This is very difficult, and typically there are sets without individual-counterparts, or individuals without set-counterparts, i.e., the selected axioms typically generate too few or too many sets. Although cumulatively generated models cannot be easily one-one correlated with a target domain, they can be seen as overall border conditions of models that are actually applied by approach (2), i.e., all applied FST models are known to be their sub-models. The type of an applied model $M$ which is one-one correlated with a target domain can be defined as the least of all cumulatively generated models whose sub-model $M$ is.

The emphasis is on (2), specifically on what Fine (2010) calls the operational approach; the alternative relational approach is discussed in the end of the section. In the operational approach, one or more axioms generate the complete structure of a set, and this set is mapped to one or more individuals in a target domain. In other words, there is initially only the target domain, such as molecule L, under scope. When we apply FST in modeling L, we first acquire an intuitive picture of L, and then we generate a FST model that matches the picture, by fixing the number of urs and the maximum rank of sets, and by selecting one or more specific construction axioms on top of extensionality and restriction.

Water molecule. The constituents of a molecule are two or more atoms which are interrelated by chemical bonds. As the types atom and molecule are different, we are dealing with vertical composition: a particular molecule is equated with a set whose members are equated with particular atoms. The constituents of a water molecule $H_{2} O$ are two hydrogen atoms $H 1$ and $H 2$ , and one oxygen atom O. $H 1$ is interrelated to O by a single covalent bond B, and $H 2$ is interrelated to O by a single covalent bond B (covalent bonds are chemical bonds). The goal is to define a minimal model that suffices for characterizing $H_{2} O$ . This model contains exactly one set ${H 1, H 2, O}$ . The minimal model $M_{3, 1}$ has three urs and the maximum rank is 1. The urs a, b, c are mapped to $H 1$ , $H 2$ , O, i.e., a, b, c are given the types H, H, O, respectively. Given the typed urs, the axiom that generates ${H 1, H 2, O}$ can be formulated as follows (see Section 3.3 for the logical form of the Γ-axioms): $\begin{matrix} Γ (r, s, t ∣ H (r) \land H (s) \land O (t) \land B (r, t) \land B (s, t)) : = {r, s, t} . \end{matrix}$

Intermolecular water composite. The constituents of an intermolecular composite are two or more molecules which are related by intermolecular bonds. As intermolecular composite and molecule are different types, we are dealing with vertical composition: a particular intermolecular composite is equated with a set whose members are equated with particular molecules. The constituents of intermolecular composite W are two water molecules $H_{2} O 1$ and $H_{2} O 2$ which are related by a hydrogen bond S. The goal is to define a minimal model that suffices for characterizing W. This model contains exactly one set ${H_{2} O 1, H_{2} O 2}$ . The minimal model $M_{2, 1}$ contains two urs and the maximum rank is 1. The urs a, b are mapped to $H_{2} O 1$ and $H_{2} O 2$ , respectively. The axiom that generates ${H_{2} O 1, H_{2} O 2}$ can be formulated as: $\begin{matrix} Γ (r, s ∣ H_{2} O (r) \land H_{2} O (s) \land S (r, s)) : = {r, s} . \end{matrix}$

Modularity I. Above, the nested structure of W was equated with the rank 1 set ${H_{2} O 1, H_{2} O 2}$ which exists in $M_{2, 1}$ , and the nested structure of $H_{2} O$ with the rank 1 set ${H 1, H 2, O}$ which exists in $M_{3, 1}$ . Now the task is to characterize the nested structure of W in both the first and the second partition levels of W. The overall model needed in capturing the first and second partitions of W is $M_{6, 2}$ . We start with 6 urs $a, b, c, d, e, f$ , give four of them the type H and two of them the type O, and enumerate them as $H 1, H 2, O 1, H 3, H 4, O 2$ for convenience. The goal is to generate the set ${{H 1, H 2, O 1}, {H 3, H 4, O 2}}$ which exists in $M_{6, 2}$ . The sets ${H 1, H 2, O 1}$ and ${H 3, H 4, O 2}$ are first generated by the axiom for water molecule and thereby typed as $H_{2} O$ . After this ${{H 1, H 2, O 1}, {H 3, H 4, O 2}}$ is generated by the axiom for intermolecular water composite, and mapped to W. W has rank 2, the $H_{2} O$ molecules have rank 1, and the H and O atoms have rank 0.

Modularity II: Types and tokens. The initiative of modeling two partition levels of W can be carried out in another more modular way, by linking two separate models. The linking of models is an application of the dichtomy of types and tokens. A certain element of one model is considered as a type, and certain elements of other models are considered as tokens (or instances) of that type. By this method, elements of one model can be applied elegantly as nested modules in other models. There is nothing pequliar about this method: it has been standard practice in ontological modeling since the antiquity, and it is standard practice in mathematical logic4

⁴
E.g. Kanamori (1994, p. 34) declares that all elements of a certain set in one model are urs of another model.

and in object-oriented programming. The type-token dichtomy is applied also in Section 4.

The exact way of implementing the type-token dichotomy is decided by the ontologist. Consider one way of modeling W by applying types and tokens. Again, the nested structure of W is equated with the rank 1 set ${H_{2} O 1, H_{2} O 2}$ which exists in $M_{2, 1}$ , and the nested structure of $H_{2} O$ is equated with the rank 1 set ${H 1, H 2, O}$ which exists in $M_{3, 1}$ . Now, the two urs ( $H_{2} O 1$ and $H_{2} O 2$ ) of $M_{2, 1}$ are interpreted as tokens whose type is the set ${H 1, H 2, O}$ of $M_{3, 1}$ . Exactly the same thing can be expressed by saying that the two urs of $M_{2, 1}$ are considered as instances of set ${H 1, H 2, O}$ of $M_{3, 1}$ . If the mappings are so to speak ‘written out’, we are again dealing with $M_{6, 2}$ .

Minimal and nonminimal D-individuals. A set that is equated with a minimal type D individual is not divisible in two or more disjoint proper subsets which are themselves equated with D-individuals. For instance, the set ${H_{2} O 1, H_{2} O 2}$ is equated with a minimal intermolecular water composite, as no two disjoint subsets of ${H_{2} O 1, H_{2} O 2}$ can be equated with intermolecular water composites. A set that is equated with a nonminimal D-individual x is divisible in two or more disjoint proper subsets which are equated with D-individuals. The constituents of the nonminimal D-individual x can be referred to as subset-constituents and relations between them. Depending on the case and the ontologist’s viewpoint, the subset-constituents of x can be considered as minimal D-individuals or other nonminimal D-individuals. It is essential that whatever formal subsets of x do not qualify as subset-constituents of x: only those subsets of x qualify as subset-constituents of x which are themselves D-individuals, i.e., structures are preserved.

As an example of a nonminimal intermolecular composite, consider a liter of water Wk which consists of k (several millions) $H_{2} O$ molecules, where every $H_{2} O$ has a hydrogen bond S with at least one other $H_{2} O$ and at most with four other $H_{2} O$ s. Wk is analogized with the set ${H_{2} O 1, H_{2} O 2, \dots, H_{2} O k}$ , which exists in $M_{k, 1}$ . Wk is a nonminimal intermolecular composite as several disjoint proper subsets of Wk are intermolecular composites. When modeling Wk, the k water molecules are mapped to the k urs in $M_{k, 1}$ . The existence of Wk is implied by the following two axioms.

$Γ (r, s ∣ H_{2} O (r) \land H_{2} O (s) \land S (r, s)) : = {r, s} .$

$\cup (x, y ∣ \exists t, u (t \in x \land u \in y \land S (t, u))) : = z .$

The Γ-axiom generates pair sets of every two water molecules that are interrelated as S. The ∪-axiom fuses together certain pairs of water molecules. It states that there exists set z which contains as members all those and only those elements that are members of x or y or both, if and only if at least one of the members of x has the relation S to at least one of the members of y. The ∪ axiom is cumulative. The order of generation is not specified, but one way to characterize it is that the axiom takes first one pair A; second, it includes all pairs that are interrelated directly to A, resulting in B; third, it includes all pairs that are interrelated directly to B, resulting in C; and so forth, until all pairs have been included in set Wk. In addition to Wk, the axioms generate all those subsets of Wk which are intermolecular water composites.

Gravitational frames. The following composition principle is applied for nested gravitational frames in celestial mechanics (Suntola, 2014, pp. 71–3): all objects which orbit the same barycenter are members of the same set. Consider the Solar System. The Earth, the Moon and various smaller objects orbit the same barycenter, which is situated near the center of the Earth, but is not exactly the center of the Earth. This makes up the set $E = {Earth, \dots, Moon}$ . Likewise, Jupiter, its moons and various smaller objects orbit the same barycenter, which makes up the set $J = {Jupiter, Io, \dots, Europa}$ . Similarly for all planets and their orbiters within the Solar System. Systems such as J, E and the Sun (S) orbit the barycenter of the Solar System, which is not the center of the Sun but close to it. All objects which orbit the barycenter of the Solar System are members of the set $SS = {S, J, \dots, E}$ . Further, SS is a member of the set whose members orbit the barycenter of the Milky Way, and so on.

The above axiom left the nested structure of $SS = {S, J, \dots, E}$ partially open, for the meaning of ‘all objects which orbit the same barycenter’ leaves space for interpretation. In one interpretation, all compact mass objects including the Earth and the moons of Jupiter are members of SS, for they orbit SS. In this model, there are only transitive membership chains, for all lower-level objects are members of all higher-level sets. When applied, this mapping requires the extra interpretation that e.g. the masses of objects that appear more than once in the hierarchy are not counted more than once. In another interpretation, one mass object such as the Earth is a member of only one set E, and belongs to SS only through E, not by being a direct member of SS. In this model, there are only antitransitive membership chains, and the axiom is translated as: all systems $a_{1}, \dots, a_{n}$ which orbit barycenter b, and where none of $a_{1}, \dots, a_{n}$ orbits any barycenter of a system that orbits b, are members of the same set. Thereby, e.g. E is a member of SS, for E orbits the barycenter of SS and does not orbit any barycenter of a system that orbits the barycenter of SS; but the Earth is not a member of SS, for the Earth orbits the barycenter of system E that orbits SS.

Social constructions. Keet and Artale (2008, Section 2.2) characterize an organizational hierarchy: “Del Piero…is member of the Juventus football team, and he is also member of the Juventus Torino club, which in turn is member of the Italian football clubs federation Federcalcio, but Del Piero is not a member of Federcalcio.” Supposed that no person is a member of Federcalcio, we are dealing with antitransitive membership chains only. A similar example is given by Johansson (2004, pp. 176–7) who talks about an antitransitive relation of direct organizational parthood that may hold between (i) a person and a local football club, and (ii) a local football club and a national football club, but (iii) not between a person and a national football club. Rules (i-iii) allow the following model.

Persons: a, b, c, d.

Local football club 1: ${a, b}$ .

Local football club 2: ${c, d}$ .

National football club: $N = {{a, b}, {c, d}}$ .

Only antitransitive membership chains are involved. Johansson concludes that direct organizational parthood ‘lacks transitivity’, but this is correct only when transitivity is explicitly denied, above by rule (iii). Consider a different rule (iv): direct organizational parthood may hold between a person and a national football club. Rules (i, ii, iv) allow the below model, where only $N^{'}$ is different from N:

National football club: $N^{'} = {a, b, c, {a, b}, {c, d}}$ .

The applied model is intransitive with respect to 2-chains of membership, as there are both transitive 2-chains such as $a \in {a, b} \in {a, b, c, {a, b}, {c, d}}$ , and an antitransitive 2-chain $d \in {c, d} \in {a, b, c, {a, b}, {c, d}}$ . Chains of the relation being a direct organizational part may be antitransitive and they may be transitive. Certainly, logic does not exclude the possibility that some organizations may have both persons and other organizations as members, nor that one organization may appear on several partition levels in another organization.

Unit sets. A special case of organizational parthood raises the need for unit (singleton) sets. Suppose that person a founds a local organization whose only member is himself, until more members join. This makes up ${a}$ . Single-person organizations may be needed also when the number of members temporarily drops to one, even though initially there were several members. Further, suppose that person a founds a national organization whose only members are for the time being a and ${a}$ . This makes up ${a, {a}}$ . The question of what is the ontological difference between a and ${a}$ and ${{a}}$ and ${a, {a}}$ can be answered: it depends on how these collections are interpreted. These might find application in modeling social constructions, or they can be excluded by appropriate axioms.

Two unit sets with an identical member. Person a belongs to two different organizations A and B, and is the only member of both. It seems that we have only one set ${a}$ but two different organizations. The set of all organizations where a belongs to would appear as ${{a}, {a}}$ . Such set does not exist in FST. The type-token dichotomy provides a solution. First, we generate model $M_{2, 2}$ which contains the set ${{a_{1}}, {a_{2}}}$ . Then, we generate another model $M_{1, 0}$ whose only element is an ur, and we assing the type ‘person a’ to this ur. Then we interpret $a_{1}$ and $a_{2}$ of ${{a_{1}}, {a_{2}}}$ of model $M_{2, 2}$ as instances of the only element of $M_{1, 0}$ . In other words, $a_{1}$ and $a_{2}$ are considered as tokens of the type ‘person a’. In this case one would not necessarily need a separate model in order to give $a_{1}$ and $a_{2}$ the type ‘person a’. However, modularity becomes handy when the structure of the applied type is more complex.

The relational approach. In the relational approach, relations such as membership, subset and overlap are assigned to hold between individuals in a target domain. Such assignments can be considered as partial axioms, as they give some information about the sets in the applied model, but typically leave much open. Although the assignment of relations only leaves nested structures of the applied sets partially open, it can be asked what is the minimal FST model that is sufficient for the given assignments. Consider a target domain of individuals $A, B, C$ which are interrelated as: $A \in B$ ; $B \subset C$ . The individuals and their relations can be mapped as: $A = a$ ; $B = {a}$ ; $C = {a, b}$ , i.e., the minimal model is $M_{2, 1}$ whose only sets are ${a}$ and ${a, b}$ . Further, it can be asked what is the minimal model with specific restrictions. If unit sets are discluded, the individuals and their relations can be mapped as: $A = a$ ; $B = {a, b}$ ; $C = {a, b, c}$ , i.e., the minimal model is $M_{3, 1}$ , whose only sets are ${a, b}$ and ${a, b, c}$ .

5. Incorporation of the functionality of discrete mereology

As transitive theories, mereologies (Section 5.1) are incapable of modeling nested structures. It is therefore intelligible to take FST or another intransitive theory as primary in modeling nested structures. However, also the functionality of transitive theories finds application in modeling nested structures. Two approaches of incorporating the functionality of discrete mereology (DM) are investigated. In Section 5.2, relations are defined in terms of FST which mimic DM relations. This approach incorporates an important portion of DM functionality, but not all of it. In Section 5.3, the functionality of complete DM is incorporated totally by applying DM as a separate theory on par with FST. The same functionality is had by an additional FST model that mimics a complete DM model.

5.1. Axioms and definitions of discrete mereology

DM operates with structureless aggregates such as $a b$ that consists of urs a, b, and $a b c d$ that consists of urs a, b, c, d. DM’s ⪯ and other relations defined in terms of ⪯ characterize relations between aggregates such as in $a b ⪯ a b c d$ and $a d ⪯ a b c d$ . An axiomatization of DM and some definitions are given; some definitions are prefixed by m to distinguish them from FST definitions with the same names.5

⁵
The axioms of extensionality, reflexivity and transitivity can be characterized as extensional mereology. Adding the axiom of discreteness to extensional mereology yields discrete mereology. Stanislaw Leśniewski axiomatized a version of mereology as early as 1916 (cf. Surma et al. (1992)). Straightforward axiomatizations can be found e.g. from Simons (1987, pp. 42–3), Sowa (2000, pp. 105–8), Casati and Varzi (1999, ch. 3) and Varzi (2016).

ax. extensionality:

$x = y \leftrightarrow \forall w (w ⪯ x \leftrightarrow w ⪯ y)$ .

ax. reflexivity:

$\forall x (x ⪯ x)$ .

ax. transitivity:

$\forall x, y, z (x ⪯ y ⪯ z \to x ⪯ z)$ .

def. proper part:

$x ⪯ y \land y ⋠ x$ , denoted as $x ≺ y$ .

def. ur-element:

$\exists y ∄ x (x ≺ y)$ , denoted as $u r (y)$ .

ax. discreteness:

$\forall x \exists y (u r (y) \land y ⪯ x)$ .

def. m-overlap:

$\exists z (z ⪯ x \land z ⪯ y)$ , denoted as $x ⊙ y$ .

def. m-disjointness:

$∄ z (z ⪯ x \land z ⪯ y)$ , denoted as $x ⌀ y$ .

def. m-intersection:

$\forall w ((w ⪯ x \land w ⪯ y) \leftrightarrow w ⪯ z)$ , denoted as $z = x \otimes y$ .

def. m-union:

$y ⪯ z \land x ⪯ z \land \forall w ((y ⪯ w \land x ⪯ w) \to z ⪯ w)$ , denoted as $z = x \oplus y$ .

def. m-difference:

$\forall w (w ⪯ z \leftrightarrow (w ⪯ x \land w ⋠ y))$ , denoted as $z = x ⊖ y$ .

George Boole (1815–1864) introduced Boolean algebras (BALGs) well before Stanislaw Leśniewski (1886–1939) introduced mereology in 1916. BALGs without the least element ∅ may be axiomatized that are isomorphic with mereologies. According to Uzquiano (2006, p. 314) “we know that a model of atomistic extensional mereology is a complete Boolean algebra (without a zero element).”6

⁶

For similar notions and relations between BALGs and mereology, see Eberle (1970), Clay (1984) and Simons (1987, p. 24).

Acknowledging the isomorphism is relevant in the sense that results achieved for either of two isomorphic theories naturally hold in both and may avoid redundant work. Whether BALG or mereology is applied is thereby a matter of taste: one can compare their axiomatizations and select one that seems more convenient.

5.2. Defining the functionality of discrete mereology

A large portion of the functionality of DM can be incorporated in FST by defining a relation analogous to DM’s primitive ⪯ in terms of FST’s membership. Although the identical symbol ‘⪯’ is used with DM, FST’s ⪯ may hold only between elements of a FST model, i.e., nothing is added to the applied FST models. The symbols x, y, z, w are variables for sets, u and h are variables for urs, and a, b, c, d denote individual urs of a FST model.

The basic idea is that membership and FST’s relations defined in Section 3 in terms of membership are structural, whereas FST’s ⪯ and relations defined in terms of ⪯ are structure-independent or structure-neutral. That ∈ and ⊂ are structural means that they are sensitive to nested structures of sets: when it is known that $a \in y$ holds, it is known that a is a member of y and exists in the first partition level of y; when it is known that $x \subseteq y$ holds, it is known that all members of x are members of y and exists in the first partition level of y. In contrast, ⪯ is characterized as structure-neutral because $a ⪯ y$ leaves open the partition level of a in y: $a ⪯ y$ entails that a exists on some level of y, but the specific level is not known. ⪯ is applied in talking about structural FST sets in structure-neutral way. Similarly as with ∈, symbols for urs may appear only on the left side of ⪯. Consider the definitions of ⪯:

def. ur part:

$u \in_{[1 rank (y)]} y$ , denoted as $u ⪯ y$ .

def. proper ur part:

$u ⪯ y \land \exists h (h ⪯ y \land h \neq u)$ , denoted as $u ≺ y$ .

def. set part:

$\forall u (u \in_{[1 rank (x)]} x \to u \in_{[1 rank (y)]} y)$ , denoted as $x ⪯ y$ .

def. proper set part:

$x ⪯ y \land y ⋠ x$ , denoted as $x ≺ y$ .

When $u ⪯ y$ holds, ur u exists in some level of set y, such as in $a ⪯ {b, {a, b}}$ . When $x ⪯ y$ holds, every ur in any level of x exists in some level of y, such as in ${a, b} ⪯ {b, {a, b}}$ . Accordingly, $y ⋠ x$ means that there is an ur in some level of y that is not in any level of x. By the definition of proper part, e.g. ${a, b} ≺ {{a, b}, {c, d}}$ and ${c, d} ≺ {{a, b}, {c, d}}$ hold. Given any kind of a membership hierarchy whatsoever, such as $a \in x \in y$ , also $a ⪯ x ⪯ y$ holds; given any kind of a subset hierarchy such as $x \subset y \subset z$ , also $x ⪯ y ⪯ z$ holds; given any kind of a hierarchy which is a combination of membership and subset relations such as $a \in x \subset y$ , also $a ⪯ x ⪯ y$ holds. Note that $x \subset y \to x ⪯ y$ holds whereas $x \subset y \to x ≺ y$ does not hold in all FST models, such as in the case where $x = {a, b}$ and $y = {a, b, {a}}$ . Fine (2010, p. 579) notes that also chains of relations such as $r \in v \subset y ≺ w$ may be used; such chains have now been given an axiomatic base and exact definitions.

The following translations of DM axioms into the terminology of FST show that FST’s ⪯ is congenial with DM axioms of reflexivity, transitivity and discreteness, but that DM extensionality must be modified by changing one of its biconditionals into an implication. This reminds that FST sets are structural whereas DM aggregates are structureless.

extensionality:

$x = y \leftrightarrow \forall w (w ⪯ x \leftrightarrow w ⪯ y)$ . This axiom does not hold, for x and y may be unidentical sets even if every ur in any level of x is found in some level of y and vice versa, such as when $x = {{a, b}, {c, d}}$ and $y = {{a, c}, {b, d}}$ . However, $x = y \to \forall w (w ⪯ x \leftrightarrow w ⪯ y)$ holds, for the identity of x and y implies that every ur that is found in some level of x is found in some level of y and vice versa.

reflexivity:

$\forall x (x ⪯ x)$ . Every ur that is found in some level of x is found in some level of x.

transitivity:

$\forall x, y, z (x ⪯ y ⪯ z \to x ⪯ z)$ . If every ur that is found in some level of x is found in some level of y and every ur that is found in some level of y is found in some level of z, then every ur that is found in some level of x is found in some level of z.

discreteness:

$\forall x \exists u (u ⪯ x)$ . Every set contains at least one ur in some level.

To illustrate how FST’s ⪯ can be applied as a structure-neutral relation in talking about structural sets, consider translations of examples (1–2) from Johansson (2004) into (1′–2′) where ⪯ is applied with typical FST models that are mapped to target domains as illustrated in Section 4.

A handle is a part of a door; a door is a part of a house; but the handle is not a part of the house.

A handle is a part of a door and a member of a door: handle ⪯ door; handle ∈ door. The door is a part of a house and a member of the house: door ⪯ house; door ∈ house. The handle is a part of the house but not a member of the house: handle ⪯ house; handle ∉ house.

$door = {handle, \dots} .$

$house = {door, \dots} = {{handle, \dots}, \dots} .$

A platoon is part of a company; a company is part of a battalion; but a platoon is not a part of a battalion.

A platoon is part of a company and a member of a company; a company is a part of a battalion and a member of the battalion; a platoon is a part of a battalion but not a member of a battalion.

As ⪯ has been defined, all DM relations that are defined in terms of ⪯ can be considered as FST definitions, including m-overlap, m-disjointness, m-intersection, m-union and m-difference.

def. m-overlap:

$\exists r (r ⪯ x \land r ⪯ y)$ , denoted as $x ⊙ y$ . At least one ur in some level of x is found in some level of y.

def. m-disjointness:

$∄ r (r ⪯ x \land r ⪯ y)$ , denoted as $x ⌀ y$ . No ur in any level of x is found in any level of y.

def. m-intersection:

$\forall w ((w ⪯ x \land w ⪯ y) \leftrightarrow w ⪯ z)$ , denoted as $z = x \otimes y$ . Every ur and only such ur that is found in some level of x and in some level of y, is found in some level of z.

def. m-union:

$y ⪯ z \land x ⪯ z \land \forall w ((y ⪯ w \land x ⪯ w) \to z ⪯ w)$ , denoted as $z = x \oplus y$ . Every ur and only such ur that is found in some level of x or y, is found in some level of z.

def. m-difference:

$\forall w (w ⪯ z \leftrightarrow (w ⪯ x \land w ⋠ y))$ , denoted as $z = x ⊖ y$ . Every ur and only such ur that is found in some level of x but not in any level of y, is found in some level of z.

As indicated in Section 3.2, these definitions are not interpreted as axioms. For instance, when ${a, b}$ and ${b, c}$ are the only sets in the applied model, e.g. ${a, b} \otimes {b, c} = {b}$ holds, but this only means that b is found in some level of both ${a, b}$ and ${b, c}$ , not that ${b}$ or ${{b}}$ or any other set whose only ur is b should exist in the model.

5.3. Applying discrete mereology as a separate theory

By mapping a complete DM model $D_{α}$ to a target domain of a FST model $M_{α, β}$ , we get a logical ground for talking freely about all imaginable parts of the target domain. The basic idea is that $M_{α, β}$ and $D_{α}$ have exactly the same number of urs; the urs in $M_{α, β}$ are one-one correlated with the urs of $D_{α}$ , the urs of $M_{α, β}$ and $D_{α}$ are one-one correlated with individuals in the target domain (that are selected as generic or indivisible with respect to this modeling initiative), and the urs of both models are mapped to exactly the same individuals of the target domain.

To guarantee that DM provides a ground for talking freely about all imaginable parts of a target domain, we must apply complete DM, where all permutations of α urs exist in $D_{α}$ : the sum of $a_{1}, \dots, a_{α}$ exists, and all its proper parts exist as elements of $D_{α}$ . Once the urs exist individually, this can be formally guaranteed by asserting $\forall x, y \exists z (z = x \oplus y)$ , where x, y, z denote aggregates. All permutations of urs exist in $M_{α, β}$ disregarding of which sets exist in $M_{α, β}$ , and even if there were no sets at all in $M_{α, β}$ .

It remains to be specified how DM relations between the aggregates of $D_{α}$ are interdependent with FST relations between the elements of $M_{α, β}$ . This requires interpreting how DM aggregates are correlated with FST sets. A provisional interpretation is given, where one set in $M_{α, β}$ is correlated with exactly one aggregate in $D_{α}$ , but one aggregate may be correlated with several sets. Set x and aggregate $a_{1} \dots a_{k}$ are correlated when each ur in aggregate $a_{1} \dots a_{k}$ is found in some level of x, and all urs in any level of x are in $a_{1} \dots a_{k}$ : $\begin{matrix} a_{1}, \dots, a_{k} \in_{[1 rank (x)]} x \land \forall u \in_{[1 rank (x)]} x (u = a_{1} \lor \dots \lor u = a_{k}) . \end{matrix}$

Consider how (some) elements of the FST model $M_{2, 2}$ in the left-hand column of the below table are correlated with aggregates of $D_{2}$ in the right-hand column. $\begin{matrix} M_{2, 2} & D_{2} \\ a; {a}; {{a}} & a \\ b; {b}; {{b}} & b \\ {a, b}; {{a}, {b}} & a b \end{matrix}$

As the correlation between sets and aggregates has been specified, interdependencies between DM relations and FST relations can be specified. FST’s part relation is written as $⪯_{FST}$ below. DM’s part relation is written as $⪯_{DM}$ . The function $agg (r)$ takes element r of $M_{α, β}$ as the input and gives as the output that element of $D_{α}$ which is correlated with r. The obvious dependency can now be formulated: $\begin{matrix} r ⪯_{FST} x \to agg (r) ⪯_{DM} agg (x) . \end{matrix}$

That $a ⪯_{FST} {a, b}$ holds in $M_{2, 2}$ implies that $a ⪯_{DM} a b$ holds in $D_{2}$ . That ${a, b} ⪯_{FST} {a, b}$ or ${a, b} ⪯_{FST} {{a, b}}$ holds in $M_{2, 2}$ implies that $a b ⪯_{DM} a b$ holds in $D_{2}$ .

It is notable that the job of a complete DM model $D_{α}$ could be done by an additional complete FST model $M_{α, 1}$ , where DM aggregates are replaced by level 1 FST sets, and ⪯ is replaced by ⊆. For instance, aggregates a, $a d$ and $a b c d$ of model $D_{4}$ are replaced by sets ${a}$ , ${a d}$ and ${a, b, c, d}$ of model $M_{4, 1}$ , respectively. Champollion and Krifka (2016) make a similar remark: “The properties of parthood described by CEM are very similar to those of subsethood in standard set theory.” As there is no empty set, FST’s subset functions in exactly the same way as DM’s part.

6. FST vs. alternative foundations of the layer-cake

A sufficient logical foundation of the layer-cake should provide (1) membership or an analogous relation for modeling transitive and antitransitive structural-vertical chains, (2) subset or an analogous relation for modeling transitive structural-horizontal chains, (3) an analog of the part relation of mereology for modeling transitive structure-neutral chains, and (4) an account of interdependencies of these relations. FST is sufficient, but also other solutions are available. In Section 6.1, the given approach of applying FST is contrasted to the on-going discussion about interdependencies of transitive and intransitive relations. In Section 6.2, FST is contrasted to classical set theories.

6.1. Discussion about transitivity and intransitivity

Any sufficient foundation of the layer-cake gives an account of how transitive and intransitive relations are applied together. The FST solution starts from membership; other relations are defined in terms of membership; interdependencies between the defined relations are implicit in the definitions: when it is known that $x \subset y$ holds, it is known that all members of x are members of y; when it is known that $x ⪯ y$ holds, it is known that any ur-element in any level of x is found in some level of y. If one wishes to talk freely about all permutations of parts of the target domain, one can apply DM as a separate theory on par with FST, or an additional FST model (Section 5.3). This approach is not intended as the final word about the team play of transitivity and intransitivity, but it is a coherent, practical and unificatory solution and a sufficent foundation of the layer-cake. Remarks about transitivity and intransitivity are now contrasted to and analyzed in the context of the given approach.7

⁷
In addition to the below citations, see also Casati and Varzi (1999), Johansson (2006) and Vieu (2006) for remarks about transitivity and intransitivity.

Consider Johansson’s remark:

be careful if you try to apply the transitivity axiom of binary mereology to parthood predicates found in areas outside mereology proper. Such predicates might very well be intransitive, nontransitive or fall outside the scope of any natural definition of transitivity. (Johansson, 2004, p. 180)

Johansson means that a transitive relation such as the mereological ⪯ is incapable of modeling antitransitive chains. However, when ⪯ is interpreted as structure-neutral, it is compatible with all structural relations. The interpretation of ⪯ as structure-neutral helps seeing Varzi’s notion in a new light:

One way or the other, the failure of transitivity as a general part-whole principle would appear to have important ramifications. Among other things, it would be detrimental to the claim, familiar from the philosophical literature, that the parthood relation modeled by classical mereology is truly governed by formal ontological principles, i.e., principles that are metaphysically neutral and domain-independent and, therefore, realized or exemplified by any entities whatsoever. …Parthood is transitive; ϕ-parthood – for many values of ‘ϕ’ – is not. (Varzi, 2005/2006, pp. 141,145)

When ⪯ is interpreted as structure-neutral, $r ⪯ x$ is compatible with whatever chains of combinations of ∈ and ⊆ relations from r to x. It is crucial that in the given approach ⪯ is especially and intentionally interpreted as a metaphysically neutral and domain-independent relation and subjugated to an intransitive theory. Why should one deny the definition of ⪯ in terms of FST, or the interpretation of DM as a structure-neutral theory? Varzi is correct in noting that some 2-chains of ϕ relations are antitransitive and some are not, for we can model transitive as well as antitransitive 2-chains by ϕ, by ∈ and by any intransitive relation. But he does not suggest the interpretation $r ϕ x \to r ⪯ x$ . The given approach is a fresh viewpoint also to the pluralism-monism distinction:

According to the pluralist about part-whole, there are different ways in which one object can be a part of another. …Although pluralism would appear to be the more plausible view, it is not the view that has been most widely held. The majority of philosophers currently working in metaphysics have been monists. They have supposed that there is but one (basic) way for a given object to be a part of another; and they have thought that this one way is the relation of part-whole explored in classical mereology…. (Fine, 2010, pp. 561–2)

The given approach is pluralist in the sense that there are different relations such as ∈ and ⪯, but monist or unificatory in the sense that ∈ is taken as primary and other relations are defined in terms of it. In the given approach, ⪯ can be considered as ‘basic parthood’ when thought of as the least committing and structure-neutral relation which does not exclude any more specific relations; but again, in FST ∈ is actually the basic relation, which is not the position in majority-monism. The given approach replaces both of the following alternatives:

We can – and this is the common strategy among mereologists currently – begin with a transitive part-relation and restrict its transitivity for certain specifications and domains.8

⁸

Seibt refers to Simons (1987), Simons (2013) and Varzi (2005/2006).

Alternatively, we can operate with a non-transitive part-relation and introduce transitivity for certain specifications and domains…. (Seibt, 2015, p. 177)

The given approach starts from ∈, but other relations are not introduced for certain specifications and domains. Instead, they are defined in terms of ∈ and applied whenever needed, as means of talking about particular ∈ relations that hold between elements of an applied model.

Seibt (2015, p. 169) acknowledges that mereology can be dropped as a primary collection theory in ontological modeling, for intransitive theories are needed to correctly characterize nested structures. Seibt considers mereology as a separate theory whose scope is diminished into characterizing only spatial regions where structural objects reside. To illustrate, given that the structure of a house is modeled by applying the intransitive relation spatial part of an object in defining antitransitive chains, it becomes intelligible to say that a certain screw S that fixes a cable running inside one of the walls of a house is not a spatial part of the house: “the wall is a spatial part of my house but something spatially in the wall apparently is not” (ibid, p. 169). Yet, as we may talk about spatial parts of structureless space where the house resides, and where spatial part of structureless space is a transitive relation, the following statement is true: “The spatial region occupied by S is a spatial part of the region occupied by this house” (ibid, p. 169). In the vocabulary of FST, all 1-members and subsets (which exist in the applied model) of the set mapped to the house are spatial parts of the house, but the screw S as a 2-member of the house is not a spatial part of the house as the membership chain is antitransitive. Applying mereology in talking about space where objects reside is compatible with applying mereology in talking about structureless relations between objects.

Finally, it is emphasised that if one starts formulating a collection theory from an intransitive basic relation, it is inevitable that this relation resembles membership. An example is given which shows that even when one does not start the axiomatization of an intransitive theory from classical set theoretic extensionality, one still ends up with a relation that resembles membership and satisfies set theoretic extensionality. Seibt (2015) starts the axiomatization of Leveled Mereology (LEM) with the below three axioms, which characterize LEM’s primitive intransitive relation ⊳. It is proved that these axioms are theorems of complete FST, which shows that ⊳ and ∈ are very similar.9

⁹

LEM is designed for modeling processes, not unchanging structures, and LEM’s identity conditions are different from FST and classical set theories. Because of these reasons, LEM is not evaluated together with FST, KPU and ZF in Section 6.2.

lem axiom of intransitivity:

$\neg (x ⊳ y ⊳ z \to x ⊳ z)$ .

lem axiom of asymmetricity:

$x ⊳ y \to \neg (y ⊳ x)$ .

lem axiom of irreflexivity:

$\neg (x ⊳ x)$ .

fst theorem of intransitivity:

$\neg \forall r, y, z (r \in y \in z \to r \in z)$ .

fst theorem of asymmetricity:

$\forall x, y (x \in y \to \neg (y \in x))$ .

fst theorem of irreflexivity:

$\forall x \neg (x \in x)$ .

FST’s theorem of intransitivity states that it is not the case that all 2-chains of membership of a model are transitive. As a proof, it suffices to show that the axioms of complete FST produce antitransitive 2-chains, starting from $M_{1, 2}$ : the chain $a \in {a} \in {{a}}$ is antitransitive; the chain $a \in {a} \in {a, {a}}$ is transitive. Asymmetricity and irreflexivity of ∈ follow from the wellfoundedness of FST models, which in turn follows from the limitation to finite rank.

6.2. FST vs. classical set theories

E.g. Zermelo–Fraenkel set theory (ZF, Jech (1978)) and Kripke–Platek set theory with ur-elements (KPU, Barwise (1975)) are sufficient foundations of the layer-cake, for all needed relations can be defined in terms of their basic intransitive membership relation, exactly as in FST. So, how to select between ZF, KPU and FST? The following interrelated notions are emphasised in reviewing them: FST is more feasible as a logical foundation of the layer-cake than classical set theories; the functionality of FST in this task is by nature programmable, whereas it is unnecessarily difficult to program the functionality of classical set theories in this task. Because of these reasons, it is wrongheaded to try to force FST into the mold of classical set theories, and vice versa. We are not just selecting between classical set theories and FST, but also between the classical set theoretic thinking and the proposed way of thinking of how FST is applied. Their difference is remarkable and it results from their entirely different intended functions. FST has been designed for modeling finite nested structures of finitely many individuals of a target domain, whereas classical set theories are intended to function as logical foundations of mathematics; it is therefore not surprising that FST and classical set theories each function fluently where they are intended, but not in each other’s domain.

In contrast to FST, the cumulative axioms of ZF and KPU define transfinite models which contain infinitely many wellfounded finite and transfinite10

¹⁰
A transfinite set consist of an infinite number of members or has a transfinite rank or both.

sets. Some transfinite sets are thought to be one-one correlated with natural numbers, rational numbers, real numbers, etc. and thereby function as logical foundations of the number classes. Such sets are not required in modeling finite structures that are dealt with e.g. in physics, cosmology, chemistry and biology. Therefore, if classical set theory is used in modeling such structures, all unnecessary sets must in any case be considered as mere technical implications of its axioms. This becomes unavoidable in computable applications where the number of actually existing elements is limited. Moreover, it is plain that not even a single unnecessary set is needed in an applied model. Therefore, the question is only about just how it is specified which elements belong to the applied model. In FST, this is done by selecting appropriate axioms. One can ask what would the axioms of such modifications of KPU and ZF look like, which generate e.g. the set {a,b} and no other sets. Some similarity with FST’s axioms is to be expected. Moreover FST is in fact just such modification of KPU that suits for modeling finite nested structures.11

¹¹

KPU and mereology were the main exemplars of FST. Disclusion of the empty set from FST was motivated by the disclusion of the least element from mereology. FST inherits sets which contain urs from KPU, but discludes the empty set and transfinite sets. Some FST definitions are modifications of KPU definitions; the modifications are needed mainly because FST discludes the empty set. FST’s axiom of extensionality is identical to extensionality in KPU. See Davis (1990, p. 49) for other versions of set theories with urs.

ZF and KPU could be applied in modeling finite individuals by the way of exclusion from a standard model. Their axioms, definitions and models could be sustained, and the applied model would be specified by excluding sets that are not needed from a transfinite standard model. For instance, suppose that we start from a standard KPU model with the goal to have an applied model whose only set is ${a, b}$ . One should first get a picture of the standard KPU model where one excludes from: its rank 0 elements are infinitely many urs $a, b, c, \dots$ and the empty set {}; its rank 1 elements are sets that are permutations of rank 0 elements; its rank 2 elements are sets that are permutations of rank 1 and rank 0 elements; ad infimum. Given the standard model, we can approach ${a, b}$ by exclusion rules (E1-2). (E1) Exclude the empty set {}, all sets with rank greater than 1, all sets which contain more than 2 elements and less than 2 elements, and all sets which contain {}. The resulting model contains a transfinite number of sets which consist of exactly two urs. (E2) select an arbitrary set from the resulting model. The selected set is the only element of the applied model.

The method of exclusion would sustain classical set theoretic thinking, but would be counter-intuitive and unnecessarily difficult for an average engineer who is not immersed in set theoretic tradition. The engineer should learn set theory in order to know what should be excluded, which requires time and efforts; it is much easier to grasp that you can generate ${a, b}$ by nesting a and b. The method of exclusion is not the common way of how the building of an entity is generally thought of, nor how the structure of an entity is generally conceived. One can start the building of a totality from elementary building blocks and start decomposing a totality into its building blocks, and one can conceive the structure of an entity by conceiveing it as a totality that is composed of interrelated bottom-level members. But it is plainly unnatural to ‘generate’ entities by excluding unnecessary elements from a transfinite space that follows from a theory which is not intended for the purpose of modeling finite nested structures.

FST is congenial with actual programming whereas classical set theory is not. FST’s nesting-axioms can be implemented simply as the function $Γ (r_{1}, r_{2}, \dots, r_{n} ∣ R) : = {r_{1}, r_{2}, \dots, r_{n}}$ which takes 1 to n inputs and returns the set that nests the inputs, given that the inputs satisfy properties and relations R. For instance, we get ${a, b}$ by assigning $α = 2$ and $β = 1$ and giving the two urs as inputs to $Γ (r_{1}, r_{2} ∣ rank (r_{1}) < β \land rank (r_{2}) < β)$ . This is in great contrast with the method of exclusion from a classical KPU model. Finite computer memory renders it impossible to actually have a transfinite model, i.e., we must select some finite rank and cardinality in any case, even though the abstract KPU model is transfinite. Further, there is no sense in actually ‘filling’ the memory of a computer by generating the greatest KPU model that fits in the memory and then excluding all unnecessary elements. Therefore, the programmer must in practice select some suitable order in which KPU axioms are applied in generating sets as long as the wanted set ${a, b}$ is generated. Even this approach requires excluding unnecessary sets. Consider the construction of set ${a, b, c}$ in steps (i–iv). (i) Let exactly 3 urs exist. (ii) Given urs a, b, c, KPU’s axiom of pairing gives sets ${a, b}$ and ${c}$ . (iii) Given ${a, b}$ and ${c}$ , the axiom of pairing gives set ${{a, b}, {c}}$ . (iv) KPU’s axiom of union gives the set that contains all members of members of a set; the existence of ${a, b, c}$ is implied by the existence of ${{a, b}, {c}}$ , as the members of members of ${{a, b}, {c}}$ are members of ${a, b, c}$ . This method of constructing sets is needed for the sake of the existence of sequences of ordinal numbers, but it is unnecessarily complex for modeling finite structures, for rank $n + 1$ sets have to be used in building rank n sets, if only KPU’s union and pairing are used in building sets.

Also the classical convention of applying sets that are produced by the axioms of KPU or ZF as grounds for the concepts of rank and cardinality functions as an abstract convention, but it is hard to see how this could help in modeling finite structures. For instance, 0 is equated with (or encoded as) set {}, 1 with set {0}, 2 with set ${0, 1}$ , 3 with set ${0, 1, 2}$ , and so on. The cumulative hierarchy of sets is revealed by replacing numbers by sets. Every set whose rank is 2 has the same rank as the set which is equated with 2; every set whose cardinality is 2 has the same cardinality as the set which is equated with 2. Similarly for all ranks and cardinalities. Classical set theories do not thereby need other axioms for natural numbers, nor FST’s recursive formulas for rank and cardinality. In contrast, FST does not provide a logical foundation for natural numbers but applies them, as the conception of natural numbers is implicit in the conceptions of rank and cardinality. Therefore, an engineer who applies FST must have a pre-theoretic intuition about natural numbers, which is exactified e.g. by Peano’s axioms. In FST, rank and cardinality of a set is calculated by applying recursive functions which actually check the contents of a set. This is obviously programmable; nothing prevents a programmer from attaching rank and cardinality as properties of a set-variable, so that one does not need to constantly re-calculate them.

All that can be done with KPU can be done with ZF, but when we descend from purely abstract constructions down to ontological modeling and programming, applying ZF as the background theory requires a further layer of encodings. The basic version of ZF comes without urs, i.e., its models consist of sets only. The empty set {} is the only indivisible entity in ZF, and every other set contains {} as the ∈-minimal rank 0 element on the bottom. If {} is mapped to individual p in the target domain, it follows that all individuals contain p, which is very often counter-intuitive. To avoid unwanted elements, one should interpret {} away as a mere technical entity. Likewise with other sets such as {{}}: one should somehow ascertain that there are no unwanted elements. A classical set theorist could handle the situation by bringing in ur-elements by encoding or equating them with certain sets, similarly as natural numbers were encoded as sets: one ur is encoded as {}, another as {{}}, another as {{}, {{}}}, and so on. This can certainly be done as an abstract procedure, but by having urs as primitives, the engineer is saved from doing such encodings. In this sense, FST and KPU suit better for ontological modeling than ZF or another set theory without urs.

In KPU, the empty set exists also as a member of sets in addition to urs. Although {} could be used similarly as an ur in KPU, one may ask what is the reason to use sets such as ${a, b, {}}$ , when one can manage with sets such as ${a, b, c}$ . Then again, one could use sub-models of basic KPU models that do not include sets which contain {}; such models would be closer to FST models. In ZF and KPU, {} is a subset of every set and the intersection of every two disjoint sets. {} could be interpreted merely as a technical marker of disjointness, but in FST this interpretation does not have to be made. Then again, one can argue that having {} as a marker of disjointness is highly practical; even so, it is still confusing to have a marker or disjointness as an element of a set.

7. Conclusions

FST incorporates only those features of classical set theories that are programmable and applicable in modeling finite nested structures, and avoids their difficulties by discluding those which are not needed. Namely, ur-elements and sets are incorporated, but transfinite sets, the empty set and all sets and encodings that are unnecessary in modeling finite nested structures are excluded. Disclusion of the unnecessary elements resulted in simpler models and very straightforward set construction. Disclusion of the empty set was motivated by the disclusion of the least element in mereology, but as an intransitive theory FST exceeds the expressive power of transitive mereology. FST’s basic structure of ur-elements nested by sets lies nearest at hand when searching a foundation for the layer-cake. FST starts from the intransitive membership relation. Other relations are defined in terms of membership, which yields their interdependencies.

The intransitive membership relation is used in modeling transitive and antitransitive structural-vertical chains, vertical chains that are combinations of transitive and antitransitive chains, and in defining other relations such as subset and part.

The transitive subset relation is used in modeling transitive structural-horizontal chains.

The transitive part relation which mimics the part relation of discrete mereology is used for talking about nested structures in a structure-neutral way.

FST’s analog of the part relation of discrete mereology (DM) left FST short of the full DM functionality, which can be had by applying DM as a separate theory on par with FST, or by applying an additional FST model which mimics a complete DM model. Any viable alternative of FST – a single theory or a group of theories – should provide analogous relations and fix their interdependencies.

FST is not the final word about collection theories in ontological modeling, but it advances the current state of the art. FST’s greatest merit is that it succeeds in its intended purpose as a unified, easily adjustable and programmable logical foundation of the layer-cake. FST exemplifies how transitive and intransitive relations can be made commensurable and applied together in two ways: by one theory where we start from an intransitive relation and define transitive relations; by two theories where we map their models and relations.

Footnotes

Acknowledgements

The creation of FST started as a reflection to John Sowa’s () critique of set theory in ontological modeling. This article is edited by A. Styrman. FST was axiomatized by A. Halko and A. Styrman. The given version of discrete mereology was axiomatized by A. Halko. We thank Torsten Hahmann and four anonymous referees of AO for extensive remarks and suggestions, which helped making this article remarkably better. We thank Ari Lehto, Heikki Sipilä and Heikki Tuononen for inspecting the examples from chemistry and Tuomo Suntola for inspecting the example about gravitational frames.

Axioms of complete FST and FST definitions

The following axioms and the assigned $α ⩾ 1$ and $β ⩾ 0$ define a complete FST model $M_{α, β}$ . Symbols x, y, z, v, w denote sets; r, s, t may denote both sets and urs; u and h denote urs.

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