Finitist set theory (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. 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.
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| - Finitist set theory (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. 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. (en)
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| - Finitist set theory (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. 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 (urs) are indivisibleprimitives. 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 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 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 constructionaxioms 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 (en)
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