Finitist set theory


Finitist set theory 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, 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 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 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 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

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 and the axiom of union of sets is a provisional union-axiom.
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.
The recursive formula gives the number of sets in :
In there are sets.
In there are 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, 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.










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 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.
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:
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 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 aggregates are structureless.
To illustrate how FST's can be applied as a structure-neutral relation in talking about structural sets, consider translations of examples where only mereology is applied, into 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.
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.