Finitist set theory

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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 , which is abbreviated as . is the collection of ur-elements of model . Ur-elements (urs) are indivisible primitives. By assigning a finite integer such as 2 as the value of , it is determined that contains exactly 2 urs. is a collection whose elements will be called sets. is a finite integer which denotes the maximum rank (nesting level) of sets in . Every set in has one or more sets or urs or both as members. The assigned and and the applied axioms fix the contents of and . To facilitate the use of language, expressions such as "sets that are elements of of model and urs that are elements of of model " are abbreviated as "sets and urs that are elements of ".

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. 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 are postulated in all versions of FST, but set construction axioms (nesting- axioms and union-axioms) vary; the assignment of finite integer values to and 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 , 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. In the axioms and definitions, symbols are variables for sets, are variables for both sets and urs, is a variable for urs, and 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 .

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 do not otherwise exclude sets such as

Complete FST models[edit]

Complete FST models contain all permutations of sets and urs within the limits of and . 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 (-axiom) and the axiom of union of sets is a provisional union-axiom (-axiom).

  • Ax. Extensionality: . Set is identical to set iff (if and only if) and have the identical members, may these be sets, urs or both.
  • Ax. Restriction: . 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: . For every ur and set that has a rank smaller than , there exists the singleton set . The rank restriction () in the axiom does the job of the axiom of foundation of traditional set theories: constraining the rank of sets to an assigned finite entails that there are no non-wellfounded sets, for such sets would have a transfinite rank. Given urs and in , the axiom of singleton sets generates only sets and , whereas the axiom of pairing of traditional set theories generates , and .
  • Ax. Union of Sets: . For all sets and , there exists set which contains as members all those and only those sets and urs that are members of , members of , or members of both and . For instance, if sets and exist, the axiom of union of sets states that the set exists. If sets and exist, the axiom states that exists. If and exist, the axiom states that exists. This axiom is different from the axiom of union of traditional set theories.[3]

Complete FST models contain all permutations of sets and urs within limits of the assigned and . The cardinality of is its number of sets and urs . Consider some examples.

: One ur exists.
: Two urs exist.
: One ur and the set exist.
: Two urs and sets , , exist.
: One ur and sets , , exist.

The recursive formula gives the number of sets in :

In there are sets.

In there are sets.

FST definitions[edit]

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 and . To illustrate a seeming conflict, suppose that and are the only sets of the applied model. The definition of intersection states that . As does not exist in the applied model, the definition of intersection may appear to be an axiom. However, this is only apparent, for does not have to exist in order to state that the only common element of and is , 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 is , is written as , and abbreviated as 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 is defined as the greatest nesting level of all -minimal elements of . The rank of is 1, as the nesting level of in is 1. The rank of is 2, as is nested by two concentric sets. The rank of is 2, as 2 is the greatest nesting level of all -minimal elements of . The rank of is 3, the rank of is 4, and so on. Formally:
is an ur-element.
, where , is defined as:
, where , is defined as:
By applying the definition of -member (below) rank can be defined as:
is an ur-element.
, is defined as
  • Def. Subset: , is denoted as . is a subset of iff every member of is a member of . Examples: ; . That is not a subset of is written as . Examples: ; . Due to the exclusion of the empty set , in FST means that all members of are members of , and there exists at least one member in and at least one member in . In traditional set theories where exist, means that does not have any members that are {\it not} members of . Therefore in traditional set theories, holds for every .
  • Def. Proper Subset: is denoted as . is a proper subset of iff is a subset of and is not a subset of . Examples: ; . That is not a proper subset of is written as . Examples: ; . In FST, means that all members of are members of , there exists at least one member in , at least two members in , and at least one member of is not a member of . In traditional set theories, means that does not have any members that are {\it not} members of , and has at least one member that is not a member of . Therefore in traditional set theories, holds for every where .
  • Def. Disjointness: is denoted as . and are disjoint iff they do not have any common members. Examples: (when ); .
  • Def. Overlap: is denoted as . and overlap iff they have one or more common members. Examples: ; . Disjointness is the contrary of overlap: ; .
  • Def. Intersection: is denoted as . The intersection of and , , contains those and only those sets and ur-elements that are members of both and . Examples: ; . As the empty set does not exist in FST, the intersection of two disjoint sets does not exist. When , is not true for any . In this case, the disjointness relation can be used: . In traditional set theories the intersection of two disjoint sets {\it is} the empty set: . 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: is denoted as . Set contains as members all those sets and ur-elements that are members of , members of , or members of both and . Examples: ; ; .
  • Theorem of Weak Supplementation: [4] Weak supplementation (WS) expresses that a proper subset of is not the whole , but must be supplemented by another subset to compose , where and are disjoint. In FST, when is a proper subset of , then has another subset that is disjoint with . For instance, is true in all FST models which contain the set .
  • Def. Difference: is denoted as The difference of and contains every member of that is not a member of . Examples: ; . As the empty set does not exist, it cannot be stated that . If is a subset of , there does not exist such that :
  • 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 is 1, disregarding whether 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. means that the cardinality of set is . E.g. , , , and .
is defined as:
, where , is defined as:
, where is defined as:
  • Def. Power Set: denoted as . Examples: ; . Power sets in FST do not contain the empty set, and thus . 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 , which makes .
  • Def. n-Member and Partition Level:
is defined as .
is defined as .
, where , is defined as .
That holds can be stated by saying that exists in the first partition level of . That holds can be stated by saying that exists in the second partition level of . And so forth. [5]
  • Def. Members.
, where , is defined as: .
is an -to- member of when is an n-member of or an n+1-member of or \ldots or an -member of .
  • Def. Partition Set. A partition set that contains all -members of a set is defined as:
.
is defined as: .
is defined as: .
  • Def. Transitive Closure: , denoted as . means that set is the transitive closure of set . contains all sets and ur-elements of the input set , i.e., the whole inner structure of . Examples:


Definitions that incorporate the functionality of discrete mereology[edit]

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 that consists of urs , and that consists of urs . DM's and other relations defined in terms of characterize relations between aggregates such as in and . An axiomatization of DM and some definitions are given; some definitions are prefixed by to distinguish them from FST's definitions with the same names.

  • ax. extensionality .
  • ax. reflexivity
  • ax. transitivity:
  • def. proper part: denoted as .
  • def. ur-element: denoted as .
  • ax. discreteness:
  • def. m-overlap: denoted as .
  • def. m-disjointness: denoted as Ø .
  • def. m-intersection: denoted as .
  • def. m-union: denoted as .
  • def. m-difference: denoted as .

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 and the goal is to mimic DM functionality, FST's may hold only between elements of a FST model, i.e., nothing is added to the applied FST models. As always, variables denote FST sets and 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 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 holds it is known that is a member of and exists in the first partition level of , and when it is known that holds it is known that all members of are members of and exists in the first partition level of . In contrast, when it is known e.g. that holds, it is not known on which specific level of does exist. is characterized as structure-neutral because it allows existing in whatever partition level of . is applied in talking about structural FST sets in a structure-neutral way. Similarly as with , symbols for urs may appear only the left side of . Consider the definitions:

  • def. ur part: denoted as .
  • def. part: , denoted as .
  • def. proper part: denoted as .

When holds, exists in some level of set . For instance, holds. When holds, every ur in any level of exists in some level of . For instance, holds. Accordingly, means that there is an ur in some level of that is not in any level of . By the definition of proper part, e.g. and hold. Given any kind of a membership hierarchy whatsoever, such as , also holds; given any kind of a subset hierarchy such as , also holds; given any kind of a hierarchy which is a combination of membership and subset relations such as , also holds. Note that holds whereas does not hold in all FST models, such as in the case where and . Fine (2010, p. 579) notes that also chains of relations such as 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 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 aggretates are structureless.

  • Extensionality: . This axiom does not hold, for and may be unidentical sets even if every ur in any level of is found in some level of and vice versa, such as when and . However, holds, for the identity of and implies that every ur that is found in some level of is found in some level of and vice versa.
  • reflexivity: Every ur that is found in some level of is found in some level of .
  • transitivity: If every ur that is found in some level of is found in some level of and every ur that is found in some level of is found in some level of , then every ur that is found in some level of is found in some level of .
  • discreteness: 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) where only mereology is applied, into (1'-2') where FST's 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 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.:
:

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 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: denoted as . At least one ur in some level of is found in some level of .
  • Def. m-disjointness: denoted as Ø . No ur in any level of is found in any level of .
  • Def. m-intersection: denoted as . is the set of all urs that are found in some levels of both and .
  • Def. m-union: denoted as . is the set of all urs in any level of or or both.
  • Def. m-difference: denoted as . is the set of all urs that are in some level of but not in any level of .

Regarding the definitions of -intersection, -union and -difference, in complete FST models all sets exist. In some incomplete FST models some do not exist. For instance, when and are the only sets in the applied model, the definition of -intersection states that , 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 is found in some level of both and .

Notes[edit]

  1. ^ Avril Styrman and Aapo Halko (2018) "Finitist set theory in ontological modeling." Applied Ontology, vol. 13, no. 2, pp. 107-133, 2018. doi:10.3233/AO-180196.
  2. ^ Wimsatt, W.C. (2006). The ontology of complex systems: Levels of organization, perspectives, and causal thickets. Canadian Journal of Philosophy, Supplementary, 20, 207–274. Fine, K. (2010). Towards a theory of part. The Journal of Philosophy, 107(11), 559–589. doi:10.5840/jphil20101071139.
  3. ^ For instance, KPU's axiom of union gives the set that contains all members of members of a set, i.e, the existence of is implied by e.g. by the existence of , as the members of members of are members of . Although such features are applied generating ordinal numbers, they are not needed in modelling finite nested structures.
  4. ^ Varzi, A.C. (2016). Mereology. In E.N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy.
  5. ^ The term 'partition level' and the recursive definition of -member are adapted from: Seibt, J. (2015) Non-transitive parthood, leveled mereology, and the representation of emergent parts of processes. Grazer Philosophische Studien, 91(1), 165–190, pp. 178-80. Seibt, J. (2009). Forms of emergent interaction in general process theory. Synthese, 166(3), 479–512, \S{3.2}. doi:10.1007/s11229-008-9373-z.