Scott–Potter set theory
Encyclopedia
An approach to the foundations of mathematics
that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician
Dana Scott
and the philosopher George Boolos
.
Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal
and ordinal number
s, Peano arithmetic
and the other usual number system
s, and the theory of relations
.
with identity
. The ontology
includes urelements as well as sets, simply to allow the set theories described in this entry to have models
that are not purely mathematical in nature. The urelements serve no essential mathematical purpose.
Some terminology peculiar to Potter's set theory:
Creation: ∀V∃V' (V∈V' ).
Remark: There is no highest level, hence there are infinitely many levels. This axiom establishes the ontology
of levels.
Separation: An axiom schema
. For any first-order formula Φ(x) with (bound) variables ranging over the level V, the collection {x∈V : Φ(x)} is also a set. (See Axiom schema of separation.)
Remark: Given the levels established by Creation, this schema establishes the existence of sets and how to form them. It tells us that a level is a set, and all subsets, definable via first-order logic
, of levels are also sets. This schema can be seen as an extension of the background logic.
Infinity: There exists at least one limit level. (See Axiom of infinity
.)
Remark: Among the sets Separation allows, at least one is infinite. This axiom is primarily mathematical, as there is no need for the actual infinite in other human contexts, the human sensory order being necessarily finite. For mathematical purposes, the axiom "There exists an inductive set
" would suffice.
Ordinals: For each (infinite) ordinal α, there exists a corresponding level Vα.
Remark: In words, "There exists a level corresponding to each infinite ordinal." Ordinals makes possible the conventional Von Neumann definition of ordinal numbers
.
Let τ(x) be a first-order term
.
Replacement
: An axiom schema
. For any collection a, ∀x∈a[τ(x) is a set] → {τ(x) : x∈a} is a set.
Remark: If the term τ(x) is a function
(call it f(x)), and if the domain
of f is a set, then the range
of f is also a set.
Let Φ denote a first-order formula
in which any number of free variables are present. Let Φ(V) denote Φ with these free variables all quantified, with the quantified variables restricted to the level V.
Reflection: An axiom schema
. If the free variables in any instance of ∃V[Φ→Φ(V)] are universally quantified, the result is an axiom.
Remark: This schema asserts the existence of a "partial" universe, namely the level V, in which all properties Φ holding when the quantified variables range over all levels, also hold when these variables range over V only. Reflection turns Creation, Infinity, Ordinals, and Replacement into theorems (Potter 2004: §13.3).
Let A and a denote sequences of nonempty set
s, each indexed by n.
Countable Choice
: Given any sequence A, there exists a sequence a such that:
Remark. Countable Choice enables proving that any set must be one of finite or infinite.
Let B and C denote sets, and let n index the members of B, each denoted Bn.
Choice: Let the members of B be disjoint nonempty sets. Then:
implements the "iterative conception of set" by stratifying the universe of sets into a series of "levels," with the sets at a given level being the members of the sets making up the next higher level. Hence the levels form a nested and well-ordered sequence, and would form a hierarchy
if set membership were transitive
. The resulting iterative conception steers clear, in a well-motivated way, of the well-known paradox
es of Russell
, Burali-Forti
, and Cantor
. These paradoxes all result from the unrestricted use of the principle of comprehension that naive set theory
allows. Collections such as "the class of all sets" or "the class of all ordinals" include sets from all levels of the hierarchy. Given the iterative conception, such collections cannot form sets at any given level of the hierarchy and thus cannot be sets at all. The iterative conception has gradually become more accepted over time, despite an imperfect understanding of its historical origins.
Boolos's (1989) axiomatic treatment of the iterative conception is his set theory S, a two sorted first order theory involving sets and levels.
. Nevertheless, Scott's theory can be seen as an axiomatization of the iterative conception and the associated iterative hierarchy.
Scott began with an axiom he declined to name: the atomic formula
x∈y implies that y is a set. In symbols:
His axiom of Extensionality
and axiom schema
of Comprehension (Separation) are strictly analogous to their ZF
counterparts and so do not mention levels. He then invoked two axioms that do mention levels:
Restriction also implies the existence of at least one level and assures that all sets are well-founded.
Scott's final axiom, the Reflection schema
, is identical to the above existence premise bearing the same name, and likewise does duty for ZF's Infinity
and Replacement. Scott's system has the same strength as ZF.
reproduces his very easy proof thereof, one requiring no set theory axioms. Thus Potter establishes from the very outset the need for a more restricted kind of collection, namely sets, that steers clear of Russell's paradox.
ZU, like ZF, cannot be finitely axiomatized. ZU differs from ZFC in that it:
Hence ZU is equivalent to the Zermelo set theory
of 1908, namely ZFC minus Choice, Replacement, and Foundation. The remaining differences between ZU and ZFC are mainly expositional.
What is the strength of ZfU, and ZFU relative to Z
, ZF
, and ZFC?
The natural number
s are not defined as a particular set within the iterative hierarchy, but as models
of a "pure" Dedekind algebra. "Dedekind algebra" is Potter's name for a set closed under an unary injective operation, successor, whose domain
contains a unique element, zero, absent from its range
. Because all Dedekind algebras with the lowest possible birthdays are categorical
(all models are isomorphic), any such algebra can proxy for the natural numbers.
The Frege–Russell definitions of the cardinal and ordinal numbers work in Scott-Potter set theory, because the equivalence classes these definitions require are indeed sets. Thus in ZU an equivalence class of:
In ZFC, defining the cardinals and ordinals in this fashion gives rise to the Cantor
and Burali-Forti paradox
, respectively.
Although Potter (2004) devotes an entire appendix to proper classes, the strength and merits of Scott-Potter set theory relative to the well-known rivals to ZFC that admit proper classes, namely NBG
and Morse–Kelley set theory
, have yet to be explored.
Scott-Potter set theory resembles NFU
in that the latter is a recently devised axiomatic set theory admitting both urelements and sets that are not well-founded. But the urelements of NFU, unlike those of ZU, play an essential role; they and the resulting restrictions on Extensionality
make possible a proof of NFU's consistency
relative to Peano arithmetic. But nothing is known about the strength of NFU relative to Creation+Separation, NFU+Infinity relative to ZU, and of NFU+Infinity+Countable Choice relative to ZU+Countable Choice.
Unlike nearly all writing on set theory in recent decades, Potter (2004) mentions mereological fusions
. His collections are also synonymous with the "virtual sets" of Willard Quine and Richard Milton Martin
: entities arising from the free use of the principle of comprehension that can never be admitted to the universe of discourse.
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Dana Scott
Dana Scott
Dana Stewart Scott is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California...
and the philosopher George Boolos
George Boolos
George Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...
.
Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
and ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s, Peano arithmetic
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
and the other usual number system
Number system
In mathematics, a 'number system' is a set of numbers, , together with one or more operations, such as addition or multiplication....
s, and the theory of relations
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
.
Preliminaries
This section and the next follow Part I of Potter (2004) closely. The background logic is first-order logicFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
with identity
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
. The ontology
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
includes urelements as well as sets, simply to allow the set theories described in this entry to have models
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
that are not purely mathematical in nature. The urelements serve no essential mathematical purpose.
Some terminology peculiar to Potter's set theory:
- a is a collection if a={x : x∈a}. All sets are collections, but not all collections are sets.
- The accumulation of a, acc(a), is the set {x : x is a urelement or ∃b∈a (x∈b or x⊂b)}.
- If ∀U∈V(U = acc(V∩U)) then V is a history.
- A level is the accumulation of a history.
- An initial level has no other levels as members.
- A limit level is a level that is neither the initial level nor the level above any other level.
- The birthday of set a, denoted V(a), is the lowest level V such that a⊂V.
Axioms
The following three axioms define the theory ZU.Creation: ∀V∃V' (V∈V' ).
Remark: There is no highest level, hence there are infinitely many levels. This axiom establishes the ontology
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
of levels.
Separation: An axiom schema
Axiom schema
In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...
. For any first-order formula Φ(x) with (bound) variables ranging over the level V, the collection {x∈V : Φ(x)} is also a set. (See Axiom schema of separation.)
Remark: Given the levels established by Creation, this schema establishes the existence of sets and how to form them. It tells us that a level is a set, and all subsets, definable via first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
, of levels are also sets. This schema can be seen as an extension of the background logic.
Infinity: There exists at least one limit level. (See Axiom of infinity
Axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
.)
Remark: Among the sets Separation allows, at least one is infinite. This axiom is primarily mathematical, as there is no need for the actual infinite in other human contexts, the human sensory order being necessarily finite. For mathematical purposes, the axiom "There exists an inductive set
Inductive set (axiom of infinity)
In the context of the axiom of infinity, an inductive set is a set X with the property that, for every x \in X, the successor x' of x is also an element of X and the set X contains the empty set \varnothing....
" would suffice.
Further existence premises
The following statements, while in the nature of axioms, are not axioms of ZU. Instead, they assert the existence of sets satisfying a stated condition. As such, they are "existence premises," meaning the following. Let X denote any statement below. Any theorem whose proof requires X is then formulated conditionally as "If X holds, then..." Potter defines several systems using existence premises, including the following two:- ZfU =df ZU + Ordinals;
- ZFU =df Separation + Reflection.
Ordinals: For each (infinite) ordinal α, there exists a corresponding level Vα.
Remark: In words, "There exists a level corresponding to each infinite ordinal." Ordinals makes possible the conventional Von Neumann definition of ordinal numbers
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
.
Let τ(x) be a first-order term
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
.
Replacement
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set...
: An axiom schema
Axiom schema
In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...
. For any collection a, ∀x∈a[τ(x) is a set] → {τ(x) : x∈a} is a set.
Remark: If the term τ(x) is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
(call it f(x)), and if the domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
of f is a set, then the range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
of f is also a set.
Let Φ denote a first-order formula
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
in which any number of free variables are present. Let Φ(V) denote Φ with these free variables all quantified, with the quantified variables restricted to the level V.
Reflection: An axiom schema
Axiom schema
In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...
. If the free variables in any instance of ∃V[Φ→Φ(V)] are universally quantified, the result is an axiom.
Remark: This schema asserts the existence of a "partial" universe, namely the level V, in which all properties Φ holding when the quantified variables range over all levels, also hold when these variables range over V only. Reflection turns Creation, Infinity, Ordinals, and Replacement into theorems (Potter 2004: §13.3).
Let A and a denote sequences of nonempty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
s, each indexed by n.
Countable Choice
Axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory, similar to the axiom of choice. It states that any countable collection of non-empty sets must have a choice function...
: Given any sequence A, there exists a sequence a such that:
- ∀n∈ω[an∈An].
Remark. Countable Choice enables proving that any set must be one of finite or infinite.
Let B and C denote sets, and let n index the members of B, each denoted Bn.
Choice: Let the members of B be disjoint nonempty sets. Then:
- ∃C∀n[C∩Bn is a singleton].
Discussion
The Von Neumann universeVon Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
implements the "iterative conception of set" by stratifying the universe of sets into a series of "levels," with the sets at a given level being the members of the sets making up the next higher level. Hence the levels form a nested and well-ordered sequence, and would form a hierarchy
Hierarchy (mathematics)
In mathematics, a hierarchy is a preorder, i.e. an ordered set. The term is used to stress a natural hierarchical relation among the elements. In particular, it is the preferred terminology for posets whose elements are classes of objects of increasing complexity. In that case, the preorder...
if set membership were transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
. The resulting iterative conception steers clear, in a well-motivated way, of the well-known paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es of Russell
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
, Burali-Forti
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
, and Cantor
Cantor's paradox
In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite...
. These paradoxes all result from the unrestricted use of the principle of comprehension that naive set theory
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
allows. Collections such as "the class of all sets" or "the class of all ordinals" include sets from all levels of the hierarchy. Given the iterative conception, such collections cannot form sets at any given level of the hierarchy and thus cannot be sets at all. The iterative conception has gradually become more accepted over time, despite an imperfect understanding of its historical origins.
Boolos's (1989) axiomatic treatment of the iterative conception is his set theory S, a two sorted first order theory involving sets and levels.
Scott's theory
Scott (1974) did not mention the "iterative conception of set," instead proposing his theory as a natural outgrowth of the simple theory of typesType theory
In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
. Nevertheless, Scott's theory can be seen as an axiomatization of the iterative conception and the associated iterative hierarchy.
Scott began with an axiom he declined to name: the atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
x∈y implies that y is a set. In symbols:
- ∀x,y∃a[x∈y→y=a].
His axiom of Extensionality
Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
and axiom schema
Axiom schema
In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...
of Comprehension (Separation) are strictly analogous to their ZF
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
counterparts and so do not mention levels. He then invoked two axioms that do mention levels:
- Accumulation. A given level "accumulates" all members and subsets of all earlier levels. See the above definition of accumulation.
- Restriction. All collections belong to some level.
Restriction also implies the existence of at least one level and assures that all sets are well-founded.
Scott's final axiom, the Reflection schema
Axiom schema
In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...
, is identical to the above existence premise bearing the same name, and likewise does duty for ZF's Infinity
Axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
and Replacement. Scott's system has the same strength as ZF.
Potter's theory
Potter (1990, 2004) introduced the idiosyncratic terminology described earlier in this entry, and discarded or replaced all of Scott's axioms except Reflection; the result is ZU. Russell's paradox is Potter's (2004) first theorem; Russell's paradoxRussell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
reproduces his very easy proof thereof, one requiring no set theory axioms. Thus Potter establishes from the very outset the need for a more restricted kind of collection, namely sets, that steers clear of Russell's paradox.
ZU, like ZF, cannot be finitely axiomatized. ZU differs from ZFC in that it:
- Includes no axiom of extensionalityAxiom of extensionalityIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
because the usual extensionality principle follows from the definition of collection and an easy lemma; - Admits nonwellfounded sets. However Potter (2004) never invokes such sets, and no theorem in Potter would be overturned were Foundation or its equivalent added to ZU;
- Includes no equivalents of Choice or the axiom schema of Replacement.
Hence ZU is equivalent to the Zermelo set theory
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
of 1908, namely ZFC minus Choice, Replacement, and Foundation. The remaining differences between ZU and ZFC are mainly expositional.
What is the strength of ZfU, and ZFU relative to Z
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
, ZF
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
, and ZFC?
The natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s are not defined as a particular set within the iterative hierarchy, but as models
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of a "pure" Dedekind algebra. "Dedekind algebra" is Potter's name for a set closed under an unary injective operation, successor, whose domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
contains a unique element, zero, absent from its range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
. Because all Dedekind algebras with the lowest possible birthdays are categorical
Categorical
See:* Categorical imperative* Morley's categoricity theorem* Categorical data analysis* Categorical distribution* Categorical logic* Categorical syllogism* Categorical proposition* Categorization* Categorical perception* Category theory...
(all models are isomorphic), any such algebra can proxy for the natural numbers.
The Frege–Russell definitions of the cardinal and ordinal numbers work in Scott-Potter set theory, because the equivalence classes these definitions require are indeed sets. Thus in ZU an equivalence class of:
- Equinumerous sets from a common level is a cardinal numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
; - Isomorphic well-orderings, also from a common level, is an ordinal numberOrdinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
.
In ZFC, defining the cardinals and ordinals in this fashion gives rise to the Cantor
Cantor's paradox
In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite...
and Burali-Forti paradox
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
, respectively.
Although Potter (2004) devotes an entire appendix to proper classes, the strength and merits of Scott-Potter set theory relative to the well-known rivals to ZFC that admit proper classes, namely NBG
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
and Morse–Kelley set theory
Morse–Kelley set theory
In the foundation of mathematics, Morse–Kelley set theory or Kelley–Morse set theory is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory...
, have yet to be explored.
Scott-Potter set theory resembles NFU
New Foundations
In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
in that the latter is a recently devised axiomatic set theory admitting both urelements and sets that are not well-founded. But the urelements of NFU, unlike those of ZU, play an essential role; they and the resulting restrictions on Extensionality
Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
make possible a proof of NFU's consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
relative to Peano arithmetic. But nothing is known about the strength of NFU relative to Creation+Separation, NFU+Infinity relative to ZU, and of NFU+Infinity+Countable Choice relative to ZU+Countable Choice.
Unlike nearly all writing on set theory in recent decades, Potter (2004) mentions mereological fusions
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
. His collections are also synonymous with the "virtual sets" of Willard Quine and Richard Milton Martin
Richard Milton Martin
Richard Milton Martin was an American logician and analytic philosopher. In his Ph.D. thesis written under Frederic Fitch, Martin discovered virtual sets a bit before Quine, and was possibly the first non-Pole other than Joseph Henry Woodger to employ a mereological system...
: entities arising from the free use of the principle of comprehension that can never be admitted to the universe of discourse.
See also
- Foundation of mathematics
- Hierarchy (mathematics)Hierarchy (mathematics)In mathematics, a hierarchy is a preorder, i.e. an ordered set. The term is used to stress a natural hierarchical relation among the elements. In particular, it is the preferred terminology for posets whose elements are classes of objects of increasing complexity. In that case, the preorder...
- List of set theory topics
- Philosophy of mathematicsPhilosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
- S (Boolos 1989)
- Von Neumann universeVon Neumann universeIn set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
- Zermelo set theoryZermelo set theoryZermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
- ZFC