Finitist set theory

Finitist set theory (FST)^[1] is a collection theory designed for modeling finite nested structures of individuals and a variety of transitive and antitransitive chains of relations between individuals. Unlike classical set theories such as ZFC and KPU, FST is not intended to function as a foundation for mathematics, but only as a tool in ontological modeling. FST functions as the logical foundation of the classical layer-cake interpretation,^[2] and manages to incorporate a large portion of the functionality of discrete mereology. FST models are of type [math]\displaystyle{ \{U_\alpha, S_\beta, \in\} }[/math], which is abbreviated as [math]\displaystyle{ \mathcal{M}_{\alpha,\beta} }[/math]. [math]\displaystyle{ U_\alpha }[/math] is the collection of ur-elements of model [math]\displaystyle{ \mathcal{M}_{\alpha,\beta} }[/math]. Ur-elements (urs) are indivisible primitives. By assigning a finite integer such as 2 as the value of [math]\displaystyle{ \alpha }[/math], it is determined that [math]\displaystyle{ U_\alpha }[/math] contains exactly 2 urs. [math]\displaystyle{ S_\beta }[/math] is a collection whose elements will be called sets. [math]\displaystyle{ \beta\geq0 }[/math] is a finite integer which denotes the maximum rank (nesting level) of sets in [math]\displaystyle{ S_\beta }[/math]. Every set in [math]\displaystyle{ S_\beta }[/math] has one or more sets or urs or both as members. The assigned [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] and the applied axioms fix the contents of [math]\displaystyle{ U_\alpha }[/math] and [math]\displaystyle{ S_\beta }[/math]. To facilitate the use of language, expressions such as "sets that are elements of [math]\displaystyle{ S_\beta }[/math] of model [math]\displaystyle{ \mathcal{M}_{\alpha,\beta} }[/math] and urs that are elements of [math]\displaystyle{ U_\alpha }[/math] of model [math]\displaystyle{ \mathcal {M}_{\alpha,\beta} }[/math]" are abbreviated as "sets and urs that are elements of [math]\displaystyle{ \mathcal{M}_{\alpha,\beta} }[/math]".

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 to 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. 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 to one correlation. The axioms of extensionality and restriction are postulated in all versions of FST, but set construction axioms (nesting- axioms and union-axioms) vary; the assignment of finite integer values to [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] 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 [math]\displaystyle{ \mathcal{M}_{\alpha, \beta} }[/math], 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 [math]\displaystyle{ \in }[/math]. All versions of FST are first-order theories. In the axioms and definitions, symbols [math]\displaystyle{ x, y, z, v, w }[/math] are variables for sets, [math]\displaystyle{ r, s, t }[/math] are variables for both sets and urs, [math]\displaystyle{ u }[/math] is a variable for urs, and [math]\displaystyle{ a, b, c, d }[/math] denote individual urs of a model. The symbols for urs may appear only on the left side of [math]\displaystyle{ \in }[/math]. The symbols for sets may appear on both [math]\displaystyle{ u \in y }[/math].

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 [math]\displaystyle{ \{a\} }[/math] do not otherwise exclude sets such as [math]\displaystyle{ \{\{a\}\}, \{\{\{a\}\}\}, \ldots \,. }[/math]

Complete FST models

Complete FST models contain all permutations of sets and urs within the limits of [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]. 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 ([math]\displaystyle{ \Gamma }[/math]-axiom) and the axiom of union of sets is a provisional union-axiom ([math]\displaystyle{ \cup }[/math]-axiom).

• Ax. Extensionality: [math]\displaystyle{ \forall r( r \in x \leftrightarrow r \in y) \leftrightarrow x=y }[/math]. Set [math]\displaystyle{ x }[/math] is identical to set [math]\displaystyle{ y }[/math] iff (if and only if) [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] have the identical members, may these be sets, urs or both.
• Ax. Restriction: [math]\displaystyle{ \forall x \exists r(r \in x) }[/math]. Every set has either a set or an ur as a member. The empty set [math]\displaystyle{ \{\} }[/math] has no members, and therefore there exists no such thing as [math]\displaystyle{ \{\} }[/math] in FST. Urs are the only [math]\displaystyle{ \in }[/math]-minimal elements in FST. Every FST set contains at least one ur as the [math]\displaystyle{ \in }[/math]-minimal member on the bottom.
• Ax. Singleton Sets: [math]\displaystyle{ \forall r_{\lt \beta}\exists x \forall s( s = r \leftrightarrow s \in x) }[/math]. For every ur and set [math]\displaystyle{ r }[/math] that has a rank smaller than [math]\displaystyle{ \beta }[/math], there exists the singleton set [math]\displaystyle{ x=\{r\} }[/math]. The rank restriction ([math]\displaystyle{ r_{\lt \beta} }[/math]) in the axiom does the job of the axiom of foundation of traditional set theories: constraining the rank of sets to an assigned finite [math]\displaystyle{ \beta }[/math] entails that there are no non-wellfounded sets, for such sets would have a transfinite rank. Given urs [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] in [math]\displaystyle{ \mathcal{M}_{2, 1} }[/math], the axiom of singleton sets generates only sets [math]\displaystyle{ \{a\} }[/math] and [math]\displaystyle{ \{b\} }[/math], whereas the axiom of pairing of traditional set theories generates [math]\displaystyle{ \{a\} }[/math], [math]\displaystyle{ \{b\} }[/math] and [math]\displaystyle{ \{a,b\} }[/math].
• Ax. Union of Sets: [math]\displaystyle{ \forall x \forall y \exists z \forall r( (r \in x \lor r \in y) \leftrightarrow r \in z) }[/math]. For all sets [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], there exists set [math]\displaystyle{ z }[/math] which contains as members all those and only those sets and urs that are members of [math]\displaystyle{ x }[/math], members of [math]\displaystyle{ y }[/math], or members of both [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]. For instance, if sets [math]\displaystyle{ \{a\} }[/math] and [math]\displaystyle{ \{b\} }[/math] exist, the axiom of union of sets states that the set [math]\displaystyle{ \{a,b\} }[/math] exists. If sets [math]\displaystyle{ \{a,b\} }[/math] and [math]\displaystyle{ \{b,c\} }[/math] exist, the axiom states that [math]\displaystyle{ \{a,b,c\} }[/math] exists. If [math]\displaystyle{ \{a\} }[/math] and [math]\displaystyle{ \{\{b\}\} }[/math] exist, the axiom states that [math]\displaystyle{ \{a,\{b\}\} }[/math] exists. This axiom is different from the axiom of union of traditional set theories.^[3]

Complete FST models [math]\displaystyle{ \mathcal{M}_{\alpha,\beta} }[/math] contain all permutations of sets and urs within limits of the assigned [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]. The cardinality of [math]\displaystyle{ \mathcal{M}_{\alpha,\beta} }[/math] is its number of sets and urs [math]\displaystyle{ sets(\alpha,\beta) + \alpha }[/math]. Consider some examples.

[math]\displaystyle{ \mathcal{M}_{1,0} }[/math]: One ur [math]\displaystyle{ a }[/math] exists. [math]\displaystyle{ card(\mathcal{M}_{1,0}) = sets(1,0) + 1 =0+1=1 }[/math]

[math]\displaystyle{ \mathcal{M}_{2,0} }[/math]: Two urs [math]\displaystyle{ a,b }[/math] exist. [math]\displaystyle{ card(\mathcal{M}_{2,0}) = sets(2,0) + 2 =0+2=2. }[/math]

[math]\displaystyle{ \mathcal{M}_{1,1} }[/math]: One ur [math]\displaystyle{ a }[/math] and the set [math]\displaystyle{ \{a\} }[/math] exist. [math]\displaystyle{ card(\mathcal{M}_{1,0}) = sets(\alpha,\beta) + \alpha = sets(1,1) + 1 =1+1=2. }[/math]

[math]\displaystyle{ \mathcal{M}_{2,1} }[/math]: Two urs [math]\displaystyle{ a,b }[/math] and sets [math]\displaystyle{ \{a\} }[/math], [math]\displaystyle{ \{b\} }[/math], [math]\displaystyle{ \{a,b\} }[/math] exist. [math]\displaystyle{ card(\mathcal{M}_{2,1}) = sets(2,1) + 2 =3+2=5. }[/math]

[math]\displaystyle{ \mathcal{M}_{1,2} }[/math]: One ur [math]\displaystyle{ a }[/math] and sets [math]\displaystyle{ \{a\} }[/math], [math]\displaystyle{ \{\{a\}\} }[/math], [math]\displaystyle{ \{a, \{a\}\} }[/math] exist. [math]\displaystyle{ card(\mathcal{M}_{1,2}) = sets(1,2) + 1 =3+1=4. }[/math]

The recursive formula [math]\displaystyle{ sets(\alpha,\beta) }[/math] gives the number of sets in [math]\displaystyle{ \mathcal{M}_{\alpha, \beta} }[/math]:

[math]\displaystyle{ sets(\alpha,0)= 0. }[/math]

[math]\displaystyle{ sets(\alpha,1)= 2^{\alpha}-1. }[/math]

[math]\displaystyle{ sets(\alpha,\beta)= 2^{\alpha + sets(\alpha,\beta-1)}-1. }[/math]

In [math]\displaystyle{ \mathcal{M}_{2,2} }[/math] there are [math]\displaystyle{ sets(2,2) = 2^{2+sets(2,1)}-1 = 2^{2+3}-1=31 }[/math] sets.

In [math]\displaystyle{ \mathcal{M}_{2,3} }[/math] there are [math]\displaystyle{ sets(2,3) = 2^{2+31}-1 = 2^{33}-1 }[/math] sets.

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 an FST model, not definitions. In order to avoid conflicts (especially with axioms for incomplete FST models), the definitions must be subjugated to the applied axioms with the given [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]. To illustrate a seeming conflict, suppose that [math]\displaystyle{ \{a,b\} }[/math] and [math]\displaystyle{ \{b,c\} }[/math] are the only sets of the applied model. The definition of intersection states that [math]\displaystyle{ \{a,b\} \cap \{b,c\}=\{b\} }[/math]. As [math]\displaystyle{ \{b\} }[/math] does not exist in the applied model, the definition of intersection may appear to be an axiom. However, this is only apparent, for [math]\displaystyle{ \{b\} }[/math] does not have to exist in order to state that the only common element of [math]\displaystyle{ \{a,b\} }[/math] and [math]\displaystyle{ \{b,c\} }[/math] is [math]\displaystyle{ b }[/math], which is the function of the definition of intersection. Similarly with all definitions.

• Def. Rank. The rank of a set is the formal analog of the level of an individual. That the rank of set [math]\displaystyle{ x }[/math] is [math]\displaystyle{ n }[/math], is written as [math]\displaystyle{ {\rm rank}(x)=n }[/math], and abbreviated as [math]\displaystyle{ x_{\beta} }[/math] 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 [math]\displaystyle{ z }[/math] is defined as the greatest nesting level of all [math]\displaystyle{ \in }[/math]-minimal elements of [math]\displaystyle{ z }[/math]. The rank of [math]\displaystyle{ \{a\} }[/math] is 1, as the nesting level of [math]\displaystyle{ a }[/math] in [math]\displaystyle{ \{a\} }[/math] is 1. The rank of [math]\displaystyle{ \{\{a\}\} }[/math] is 2, as [math]\displaystyle{ a }[/math] is nested by two concentric sets. The rank of [math]\displaystyle{ \{\{a\},a\} }[/math] is 2, as 2 is the greatest nesting level of all [math]\displaystyle{ \in }[/math]-minimal elements of [math]\displaystyle{ \{\{a\},a\} }[/math]. The rank of [math]\displaystyle{ \{\{\{a\}\}\} }[/math] is 3, the rank of [math]\displaystyle{ \{\{\{\{a\}\}\}\} }[/math] is 4, and so on. Formally:

[math]\displaystyle{ {\rm rank}(s)=0 \leftrightarrow s }[/math] is an ur-element.

[math]\displaystyle{ {\rm rank}(s) \geq n }[/math], where [math]\displaystyle{ n \geq 1 }[/math], is defined as: [math]\displaystyle{ \exists s_{1}, s_{2}, \ldots, s_{n} (s_{1} \in s_{2} \in \ldots \in s_{n} \in s ). }[/math]

[math]\displaystyle{ {\rm rank}(s) = n }[/math], where [math]\displaystyle{ n \geq 1 }[/math], is defined as: [math]\displaystyle{ {\rm rank}(s) \geq n \land \lnot ({\rm rank}(s) \geq n+1). }[/math]

By applying the definition of [math]\displaystyle{ n }[/math]-member (below) rank can be defined as:

[math]\displaystyle{ {\rm rank}(s)=0 \leftrightarrow s }[/math] is an ur-element.

[math]\displaystyle{ {\rm rank}(r)= n }[/math], is defined as [math]\displaystyle{ \exists a (a \in_{n} x) \land \not\exists a (a \in_{n+1} x ). }[/math]

• Def. Subset: [math]\displaystyle{ \forall r( r \in x \rightarrow r \in y) }[/math], is denoted as [math]\displaystyle{ x \subseteq y }[/math]. [math]\displaystyle{ x }[/math] is a subset of [math]\displaystyle{ y }[/math] iff every member of [math]\displaystyle{ x }[/math] is a member of [math]\displaystyle{ y }[/math]. Examples: [math]\displaystyle{ \{a\} \subseteq \{a\} }[/math]; [math]\displaystyle{ \{a,b\} \subseteq \{a,b,c\} }[/math]. That [math]\displaystyle{ x }[/math] is not a subset of [math]\displaystyle{ y }[/math] is written as [math]\displaystyle{ x \not\subseteq y }[/math]. Examples: [math]\displaystyle{ \{a\} \not\subseteq \{b\} }[/math]; [math]\displaystyle{ \{a,b\} \not\subseteq \{b,c\} }[/math]. Due to the exclusion of the empty set [math]\displaystyle{ \{\} }[/math], in FST [math]\displaystyle{ x \subseteq y }[/math] means that all members of [math]\displaystyle{ x }[/math] are members of [math]\displaystyle{ y }[/math], and there exists at least one member in [math]\displaystyle{ x }[/math] and at least one member in [math]\displaystyle{ y }[/math]. In traditional set theories where [math]\displaystyle{ \{\} }[/math] exist, [math]\displaystyle{ x \subseteq y }[/math] means that [math]\displaystyle{ x }[/math] does not have any members that are {\it not} members of [math]\displaystyle{ y }[/math]. Therefore in traditional set theories, [math]\displaystyle{ \{\} \subseteq y }[/math] holds for every [math]\displaystyle{ y }[/math].

• Def. Proper Subset: [math]\displaystyle{ x \subseteq y \land y \not\subseteq x, }[/math] is denoted as [math]\displaystyle{ x \subset y }[/math]. [math]\displaystyle{ x }[/math] is a proper subset of [math]\displaystyle{ y }[/math] iff [math]\displaystyle{ x }[/math] is a subset of [math]\displaystyle{ y }[/math] and [math]\displaystyle{ y }[/math] is not a subset of [math]\displaystyle{ x }[/math]. Examples: [math]\displaystyle{ \{a\} \subset \{a,b\} }[/math]; [math]\displaystyle{ \{a,b\} \subset \{a,b,c\} }[/math]. That [math]\displaystyle{ x }[/math] is not a proper subset of [math]\displaystyle{ y }[/math] is written as [math]\displaystyle{ x \not\subset y }[/math]. Examples: [math]\displaystyle{ \{a\} \not\subset \{a\} }[/math]; [math]\displaystyle{ \{a,b,c\} \not\subset \{a,b\} }[/math]. In FST, [math]\displaystyle{ x \subset y }[/math] means that all members of [math]\displaystyle{ x }[/math] are members of [math]\displaystyle{ y }[/math], there exists at least one member in [math]\displaystyle{ x }[/math], at least two members in [math]\displaystyle{ y }[/math], and at least one member of [math]\displaystyle{ y }[/math] is not a member of [math]\displaystyle{ x }[/math]. In traditional set theories, [math]\displaystyle{ x \subset y }[/math] means that [math]\displaystyle{ x }[/math] does not have any members that are {\it not} members of [math]\displaystyle{ y }[/math], and [math]\displaystyle{ y }[/math] has at least one member that is not a member of [math]\displaystyle{ x }[/math]. Therefore in traditional set theories, [math]\displaystyle{ \{\} \subset y }[/math] holds for every [math]\displaystyle{ y }[/math] where [math]\displaystyle{ y \neq \{\} }[/math].

• Def. Disjointness: [math]\displaystyle{ \not\exists r(r \in x \land r \in y), }[/math] is denoted as [math]\displaystyle{ x \wr y }[/math]. [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are disjoint iff they do not have any common members. Examples: [math]\displaystyle{ \{a\} \wr \{b\} }[/math] (when [math]\displaystyle{ a \neq b }[/math]); [math]\displaystyle{ \{a\} \wr \{\{a\}\} }[/math].

• Def. Overlap: [math]\displaystyle{ \exists r(r \in x \land r \in y), }[/math] is denoted as [math]\displaystyle{ x \circ y }[/math]. [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] overlap iff they have one or more common members. Examples: [math]\displaystyle{ \{a\} \circ \{a\} }[/math]; [math]\displaystyle{ \{a,b\} \circ \{b,c\} }[/math]. Disjointness is the contrary of overlap: [math]\displaystyle{ x \circ y \leftrightarrow \lnot(x \wr y) }[/math]; [math]\displaystyle{ \lnot(x \circ y) \leftrightarrow x \wr y }[/math].

• Def. Intersection: [math]\displaystyle{ \forall r ( (r \in x \land r \in y) \leftrightarrow r \in z), }[/math] is denoted as [math]\displaystyle{ z=x \cap y }[/math]. The intersection of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], [math]\displaystyle{ z=x \cap y }[/math], contains those and only those sets and ur-elements that are members of both [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]. Examples: [math]\displaystyle{ \{a\} \cap \{a\} = \{a\} }[/math]; [math]\displaystyle{ \{a,b,c\} \cap \{b,c,d\} = \{b,c\} }[/math]. As the empty set does not exist in FST, the intersection of two disjoint sets does not exist. When [math]\displaystyle{ r \neq s }[/math], [math]\displaystyle{ \{r\} \cap \{s\} =y }[/math] is not true for any [math]\displaystyle{ y }[/math]. In this case, the disjointness relation [math]\displaystyle{ \wr }[/math] can be used: [math]\displaystyle{ \{r\} \wr \{s\} }[/math]. In traditional set theories the intersection of two disjoint sets {\it is} the empty set: [math]\displaystyle{ \{a\} \cap \{b\} = \{\} }[/math]. If the axiom of restriction were deleted and the existence of the empty set were postulated, this would still not imply that the empty set {\it is} the intersection of two disjoint sets.

• Def. Union: [math]\displaystyle{ \forall r ( (r \in x \lor r \in y) \leftrightarrow r \in z), }[/math] is denoted as [math]\displaystyle{ z=x \cup y }[/math]. Set [math]\displaystyle{ z }[/math] contains as members all those sets and ur-elements that are members of [math]\displaystyle{ x }[/math], members of [math]\displaystyle{ y }[/math], or members of both [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]. Examples: [math]\displaystyle{ \{a\} \cup \{a\} = \{a\} }[/math]; [math]\displaystyle{ \{a,b\} \cup \{b,c\} = \{a,b,c\} }[/math]; [math]\displaystyle{ \{a\} \cup \{\{b\}\} = \{a,\{b\}\} }[/math].

• Theorem of Weak Supplementation: [math]\displaystyle{ x \subset y \rightarrow \exists z (z \subset y \land z \wr x). }[/math] ^[4] Weak supplementation (WS) expresses that a proper subset [math]\displaystyle{ x }[/math] of [math]\displaystyle{ y }[/math] is not the whole [math]\displaystyle{ y }[/math], but must be supplemented by another subset [math]\displaystyle{ z }[/math] to compose [math]\displaystyle{ y }[/math], where [math]\displaystyle{ z }[/math] and [math]\displaystyle{ x }[/math] are disjoint. In FST, when [math]\displaystyle{ x }[/math] is a proper subset of [math]\displaystyle{ y }[/math], then [math]\displaystyle{ y }[/math] has another subset [math]\displaystyle{ z }[/math] that is disjoint with [math]\displaystyle{ x }[/math]. For instance, [math]\displaystyle{ \{a\} \subset \{a,b\} \rightarrow (\{b\} \subset \{a,b\} \land \{b\} \wr \{a\}) }[/math] is true in all FST models which contain the set [math]\displaystyle{ \{a,b\} }[/math].

• Def. Difference: [math]\displaystyle{ \forall r ( r \in z \leftrightarrow (r \in x \land r \notin y)), }[/math] is denoted as [math]\displaystyle{ z=x \setminus y. }[/math] The difference [math]\displaystyle{ z }[/math] of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] contains every member of [math]\displaystyle{ x }[/math] that is not a member of [math]\displaystyle{ y }[/math]. Examples: [math]\displaystyle{ \{a\} \setminus \{b\} =\{a\} }[/math]; [math]\displaystyle{ \{a,b,c\} \setminus \{a,b\} =\{c\} }[/math]. As the empty set does not exist, it cannot be stated that [math]\displaystyle{ \{x\} \setminus \{x\}=\{\} }[/math]. If [math]\displaystyle{ x }[/math] is a subset of [math]\displaystyle{ y }[/math], there does not exist [math]\displaystyle{ z }[/math] such that [math]\displaystyle{ z= x \setminus y }[/math]: [math]\displaystyle{ \forall x,y(x \subseteq y \leftrightarrow \not\exists z(z=x \setminus y)). }[/math]

• Def. Cardinality. Cardinality denotes the number of members of a set. Cardinality is defined only for sets: ur-elements do not have a cardinality. The cardinality of [math]\displaystyle{ \{r\} }[/math] is 1, disregarding whether [math]\displaystyle{ r }[/math] 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 [math]\displaystyle{ \{\} }[/math] is 0. [math]\displaystyle{ {\rm card}(x)=n }[/math] means that the cardinality of set [math]\displaystyle{ x }[/math] is [math]\displaystyle{ n }[/math]. E.g. [math]\displaystyle{ {\rm card}(\{y,z\})=2 }[/math], [math]\displaystyle{ {\rm card}(\{x,y,\{z\}\})=3 }[/math], [math]\displaystyle{ {\rm card}(\{ x, \{x\}, \{\{x\}\} \})=3 }[/math], and [math]\displaystyle{ {\rm card}(\{x,y,z,w\})=4 }[/math].

[math]\displaystyle{ {\rm card}(x) =1 }[/math] is defined as: [math]\displaystyle{ \exists s(s \in x \land \forall r (r \in x \leftrightarrow r=s)). }[/math]

[math]\displaystyle{ {\rm card}(x) \geq n }[/math], where [math]\displaystyle{ n \geq 2 }[/math], is defined as: [math]\displaystyle{ \exists s_{1}, s_{2}, \ldots , s_{n} ( (\bigwedge_{k=1}^{n} s_{k} \in x) \land \bigwedge_{a=1}^{n-1} \bigwedge_{b=a+1}^{n} s_{a} \neq s_{b} ). }[/math]

[math]\displaystyle{ {\rm card}(x) = n }[/math], where [math]\displaystyle{ n \geq 2 }[/math] is defined as: [math]\displaystyle{ {\rm card}(x) \geq n \land \lnot ({\rm card}(x) \geq n+1). }[/math]

• Def. Power Set: [math]\displaystyle{ \forall z (z \in y \leftrightarrow z \subseteq x), }[/math] denoted as [math]\displaystyle{ y=P(x) }[/math]. Examples: [math]\displaystyle{ P(\{r\}) = \{\{r\}\} }[/math]; [math]\displaystyle{ P(\{r,s\}) = \{ \{r\}, \{s\}, \{r,s\} \} }[/math]. Power sets in FST do not contain the empty set, and thus [math]\displaystyle{ {\rm card}(P(x))=2^{{\rm card}(x)} -1 }[/math]. 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 [math]\displaystyle{ P(\{y,x\})=\{\{\}, \{y\}, \{z\}, \{y,z\} \} }[/math], which makes [math]\displaystyle{ {\rm card}(P(x))=2^{{\rm card}(x)} }[/math].

• Def. n-Member and Partition Level:

[math]\displaystyle{ r \in_{1} x }[/math] is defined as [math]\displaystyle{ r \in x }[/math].

[math]\displaystyle{ r \in_{2} x }[/math] is defined as [math]\displaystyle{ \exists y (r \in y \in x) }[/math].

[math]\displaystyle{ r \in_{n} x }[/math], where [math]\displaystyle{ n \geq 2 }[/math], is defined as [math]\displaystyle{ \exists y (r \in_{n-1} y \in x) }[/math].

That [math]\displaystyle{ r \in_{1} x }[/math] holds can be stated by saying that [math]\displaystyle{ r }[/math] exists in the first partition level of [math]\displaystyle{ x }[/math]. That [math]\displaystyle{ r \in_{2} x }[/math] holds can be stated by saying that [math]\displaystyle{ r }[/math] exists in the second partition level of [math]\displaystyle{ x }[/math]. And so forth. ^[5]

• Def. [math]\displaystyle{ [n\, m] }[/math] Members.

[math]\displaystyle{ r \in_{[n\, m]} x }[/math], where [math]\displaystyle{ n \leq m }[/math], is defined as: [math]\displaystyle{ r \in_{n} x \lor r \in_{n+1} x \lor r \in_{n+2} x \lor \ldots \lor r \in_{m} x }[/math].

[math]\displaystyle{ r }[/math] is an [math]\displaystyle{ n }[/math]-to-[math]\displaystyle{ m }[/math] member of [math]\displaystyle{ x }[/math] when [math]\displaystyle{ r }[/math] is an n-member of [math]\displaystyle{ x }[/math] or an n+1-member of [math]\displaystyle{ x }[/math] or \ldots or an [math]\displaystyle{ m }[/math]-member of [math]\displaystyle{ x }[/math].

• Def. Partition Set. A partition set that contains all [math]\displaystyle{ n }[/math]-members of a set is defined as:

[math]\displaystyle{ partition_{1}(x)=x }[/math].

[math]\displaystyle{ partition_{2}(x)=y }[/math] is defined as: [math]\displaystyle{ \forall r ( r \in_{2} x \leftrightarrow r \in y) }[/math].

[math]\displaystyle{ partition_{n}(x)=y }[/math] is defined as: [math]\displaystyle{ \forall r ( r \in_{n} x \leftrightarrow r \in y) }[/math].

• Def. Transitive Closure: [math]\displaystyle{ \forall r ( r \in_{[1\, rank(x)]} x \leftrightarrow r \in y) }[/math], denoted as [math]\displaystyle{ y=Tc(x) }[/math]. [math]\displaystyle{ y=Tc(x) }[/math] means that set [math]\displaystyle{ y }[/math] is the transitive closure of set [math]\displaystyle{ x }[/math]. [math]\displaystyle{ y }[/math] contains all sets and ur-elements of the input set [math]\displaystyle{ x }[/math], i.e., the whole inner structure of [math]\displaystyle{ x }[/math]. Examples:

[math]\displaystyle{ Tc(\{\{a,b\}\})=\{a,b,\{a,b\}\}. }[/math]

[math]\displaystyle{ Tc(\{\{\{a\}\}\})=\{a,\{a\},\{\{a\}\}\}. }[/math]

[math]\displaystyle{ Tc(\{a,\{a\}\})=\{a,\{a\}\}. }[/math]

Definitions that incorporate the functionality of discrete mereology

As transitive theories, Mereology and Boolean algebra 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. A large portion of the functionality of discrete mereology (DM) can be incorporated in FST, in terms of relations which mimic DM's relations.

DM operates with structureless aggregates such as [math]\displaystyle{ ab }[/math] that consists of urs [math]\displaystyle{ a,b }[/math], and [math]\displaystyle{ abcd }[/math] that consists of urs [math]\displaystyle{ a,b,c,d }[/math]. DM's [math]\displaystyle{ \preceq }[/math] and other relations defined in terms of [math]\displaystyle{ \preceq }[/math] characterize relations between aggregates such as in [math]\displaystyle{ ab \preceq abcd }[/math] and [math]\displaystyle{ ad \preceq abcd }[/math]. An axiomatization of DM and some definitions are given; some definitions are prefixed by [math]\displaystyle{ m }[/math] to distinguish them from FST's definitions with the same names.

• ax. extensionality [math]\displaystyle{ x=y \leftrightarrow \forall w (w \preceq x \leftrightarrow w \preceq y) }[/math].
• ax. reflexivity [math]\displaystyle{ \forall x (x \preceq x). }[/math]
• ax. transitivity: [math]\displaystyle{ \forall x,y,z (x \preceq y \preceq z \rightarrow x \preceq z). }[/math]
• def. proper part: [math]\displaystyle{ x \preceq y \land y \not\preceq x, }[/math] denoted as [math]\displaystyle{ x \prec y }[/math].
• def. ur-element: [math]\displaystyle{ \not\exists x (x \prec y), }[/math] denoted as [math]\displaystyle{ ur(y) }[/math].
• ax. discreteness: [math]\displaystyle{ \forall x \exists y(ur(y) \land y \preceq x). }[/math]
• def. m-overlap: [math]\displaystyle{ \exists z (z \preceq x \land z \preceq y), }[/math] denoted as [math]\displaystyle{ x \odot y }[/math].
• def. m-disjointness: [math]\displaystyle{ \not\exists z (z \preceq x \land z \preceq y), }[/math] denoted as [math]\displaystyle{ x }[/math] Ø [math]\displaystyle{ y }[/math].
• def. m-intersection: [math]\displaystyle{ \forall w ( (w \preceq x \land w \preceq y) \leftrightarrow w \preceq z), }[/math] denoted as [math]\displaystyle{ z=x \otimes y }[/math].
• def. m-union: [math]\displaystyle{ y \preceq z \land x \preceq z \land \forall w ((y \preceq w \land x \preceq w) \rightarrow z \preceq w), }[/math] denoted as [math]\displaystyle{ z=x \oplus y }[/math].
• def. m-difference: [math]\displaystyle{ \forall w (w \preceq z \leftrightarrow (w \preceq x \land w \not\preceq y) ), }[/math] denoted as [math]\displaystyle{ z=x \ominus y }[/math].

A large portion of the functionality of DM can be incorporated in FST by defining a relation analogous to DM's primitive [math]\displaystyle{ \preceq }[/math] in terms of FST's membership. Although the identical symbol '[math]\displaystyle{ \preceq }[/math]' is used with DM and the goal is to mimic DM functionality, FST's [math]\displaystyle{ \preceq }[/math] may hold only between elements of a FST model, i.e., nothing is added to the applied FST models. As always, variables [math]\displaystyle{ x,y,z,w }[/math] denote FST sets and [math]\displaystyle{ u }[/math] denotes an ur-element.

The basic idea is that membership and FST's basic relations defined in terms of membership are structural, whereas FST's [math]\displaystyle{ \preceq }[/math] and relations defined in terms of [math]\displaystyle{ \preceq }[/math] are structure-independent or structure-neutral. That [math]\displaystyle{ \in }[/math] and [math]\displaystyle{ \subset }[/math] are structural means that they are sensitive to nested structures of sets: when it is known that [math]\displaystyle{ u \in y }[/math] holds it is known that [math]\displaystyle{ u }[/math] is a member of [math]\displaystyle{ y }[/math] and exists in the first partition level of [math]\displaystyle{ y }[/math], and when it is known that [math]\displaystyle{ x \subseteq y }[/math] holds it is known that all members of [math]\displaystyle{ x }[/math] are members of [math]\displaystyle{ y }[/math] and exists in the first partition level of [math]\displaystyle{ y }[/math]. In contrast, when it is known e.g. that [math]\displaystyle{ u \preceq y }[/math] holds, it is not known on which specific level of [math]\displaystyle{ y }[/math] does [math]\displaystyle{ u }[/math] exist. [math]\displaystyle{ \preceq }[/math] is characterized as structure-neutral because it allows [math]\displaystyle{ u }[/math] existing in whatever partition level of [math]\displaystyle{ y }[/math]. [math]\displaystyle{ \preceq }[/math] is applied in talking about structural FST sets in a structure-neutral way. Similarly as with [math]\displaystyle{ \in }[/math], symbols for urs may appear only the left side of [math]\displaystyle{ \preceq }[/math]. Consider the definitions:

• def. ur part: [math]\displaystyle{ u \in_{[1\,rank(y)]} y, }[/math] denoted as [math]\displaystyle{ u \preceq y }[/math].
• def. part: [math]\displaystyle{ \forall u (u \in_{[1\,rank(x)]} x \rightarrow u \in_{[1\,rank(y)]} y) }[/math], denoted as [math]\displaystyle{ x \preceq y }[/math].
• def. proper part: [math]\displaystyle{ x \preceq y \land y \not\preceq x, }[/math] denoted as [math]\displaystyle{ x \prec y }[/math].

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

The following translations of DM axioms into the terminology of FST show that FST's [math]\displaystyle{ \preceq }[/math] is congenial with DM's axioms of reflexivity, transitivity and discreteness, but that DM extensionality must be modified by changing one of its equivalence relations into an implication. This reminds that FST sets are structural whereas DM aggregates are structureless.

• Extensionality: [math]\displaystyle{ x=y \leftrightarrow \forall w (w \preceq x \leftrightarrow w \preceq y) }[/math]. This axiom does not hold, for [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] may be unidentical sets even if every ur in any level of [math]\displaystyle{ x }[/math] is found in some level of [math]\displaystyle{ y }[/math] and vice versa, such as when [math]\displaystyle{ x=\{\{a,b\},\{c,d\}\} }[/math] and [math]\displaystyle{ y = \{\{a,c\},\{b,d\}\} }[/math]. However, [math]\displaystyle{ x=y \rightarrow \forall w (w \preceq x \leftrightarrow w \preceq y) }[/math] holds, for the identity of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] implies that every ur that is found in some level of [math]\displaystyle{ x }[/math] is found in some level of [math]\displaystyle{ y }[/math] and vice versa.
• reflexivity: [math]\displaystyle{ \forall x (x \preceq x). }[/math] Every ur that is found in some level of [math]\displaystyle{ x }[/math] is found in some level of [math]\displaystyle{ x }[/math].
• transitivity: [math]\displaystyle{ \forall x,y,z (x \preceq y \preceq z \rightarrow x \preceq z). }[/math] If every ur that is found in some level of [math]\displaystyle{ x }[/math] is found in some level of [math]\displaystyle{ y }[/math] and every ur that is found in some level of [math]\displaystyle{ y }[/math] is found in some level of [math]\displaystyle{ z }[/math], then every ur that is found in some level of [math]\displaystyle{ x }[/math] is found in some level of [math]\displaystyle{ z }[/math].
• discreteness: [math]\displaystyle{ \forall x \exists u (u \preceq x). }[/math] Every set contains at least one ur in some level.

To illustrate how FST's [math]\displaystyle{ \preceq }[/math] can be applied as a structure-neutral relation in talking about structural sets, consider translations of examples (1-2) where only mereology is applied, into (1'-2') where FST's [math]\displaystyle{ \preceq }[/math] is be applied together with membership.

1. 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.:
1'. A handle is a part of a door and a member of a door: handle [math]\displaystyle{ \preceq }[/math] door; handle [math]\displaystyle{ \in }[/math] door. The door is a part of a house and a member of the house: door [math]\displaystyle{ \preceq }[/math] house; door [math]\displaystyle{ \in }[/math] house. The handle is a part of the house but not a member of the house: handle [math]\displaystyle{ \preceq }[/math] house; handle [math]\displaystyle{ \not\in }[/math] house.:
[math]\displaystyle{ door = \{handle, \ldots\}. }[/math]:
[math]\displaystyle{ house = \{door, \ldots\} = \{\{handle, \ldots\}, \ldots\}. }[/math]

2. A platoon is part of a company; a company is part of a battalion; but a platoon is not a part of a battalion.:
2'. 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 [math]\displaystyle{ \preceq }[/math] has been defined, all DM relations that are defined in terms of [math]\displaystyle{ \preceq }[/math] can be considered as FST definitions, including m-overlap, m-disjointness, m-intersection, m-union and m-difference.

• Def. m-overlap: [math]\displaystyle{ \exists r (r \preceq x \land r \preceq y), }[/math] denoted as [math]\displaystyle{ x \odot y }[/math]. At least one ur in some level of [math]\displaystyle{ x }[/math] is found in some level of [math]\displaystyle{ y }[/math].
• Def. m-disjointness: [math]\displaystyle{ \not\exists r (r \preceq x \land r \preceq y), }[/math] denoted as [math]\displaystyle{ x }[/math] Ø [math]\displaystyle{ y }[/math]. No ur in any level of [math]\displaystyle{ x }[/math] is found in any level of [math]\displaystyle{ y }[/math].
• Def. m-intersection: [math]\displaystyle{ \forall w ( (w \preceq x \land w \preceq y) \leftrightarrow w \preceq z), }[/math] denoted as [math]\displaystyle{ z=x \otimes y }[/math]. [math]\displaystyle{ z }[/math] is the set of all urs that are found in some levels of both [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math].
• Def. m-union: [math]\displaystyle{ y \preceq z \land x \preceq z \land \forall w ((y \preceq w \land x \preceq w) \rightarrow z \preceq w), }[/math] denoted as [math]\displaystyle{ z=x \oplus y }[/math]. [math]\displaystyle{ z }[/math] is the set of all urs in any level of [math]\displaystyle{ x }[/math] or [math]\displaystyle{ y }[/math] or both.
• Def. m-difference: [math]\displaystyle{ \forall w (w \preceq z \leftrightarrow (w \preceq x \land w \not\preceq y) ), }[/math] denoted as [math]\displaystyle{ z=x \ominus y }[/math]. [math]\displaystyle{ z }[/math] is the set of all urs that are in some level of [math]\displaystyle{ x }[/math] but not in any level of [math]\displaystyle{ y }[/math].

Regarding the definitions of [math]\displaystyle{ m }[/math]-intersection, [math]\displaystyle{ m }[/math]-union and [math]\displaystyle{ m }[/math]-difference, in complete FST models all sets [math]\displaystyle{ z }[/math] exist. In some incomplete FST models some [math]\displaystyle{ z }[/math] do not exist. For instance, when [math]\displaystyle{ \{a,b\} }[/math] and [math]\displaystyle{ \{b,c\} }[/math] are the only sets in the applied model, the definition of [math]\displaystyle{ m }[/math]-intersection states that [math]\displaystyle{ \{a,b\} \otimes \{b,c\}=\{b\} }[/math], which makes the definition appear as an axiom. As indicated above, the definition is not interpreted as an axiom, but only as a formula which states that [math]\displaystyle{ b }[/math] is found in some level of both [math]\displaystyle{ \{a,b\} }[/math] and [math]\displaystyle{ \{b,c\} }[/math].

Notes

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