Finitist set theory (FST) is a collection theory designed for modeling finite nested structures of individuals and a variety of
transitive and
antitransitive
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which descri ...
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-element
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual.
Theory
There ...
s 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
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s 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 compound
A chemical compound is a chemical substance composed of many identical molecules (or molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A molecule consisting of atoms of only one element ...
s or
social construction
Social constructionism is a theory in sociology, social ontology, and communication theory which proposes that certain ideas about physical reality arise from collaborative consensus, instead of pure observation of said reality. The theory ...
s 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
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
and all its sets have a finite
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
and cardinality. FST axioms are formulated by
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
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 (
-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
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
A refers to a union of ze ...
:
. 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.
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
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 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 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 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 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:
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.
[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. .]
* Def.