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\documentclass[a4,10pt]article oldlogicmacr {} {ι} set theories . Randall Holmes, Thomas Forster and Thierry Libert

The one thing that all alternative set theories have in common is the fact that they are alternatives to

We will briefly cover simple type theory and Zermelo set theory, which are not strictly speaking alternative set theories (type theory is not precisely set theory and Zermelo set theory is not precisely “alternative”, as it is the original version of

The next section takes us farther from the familiar. New Foundations and the known-to-be consistent theories allied with it are motivated by simple type theory rather than Zermelo set theory and superficially look very different from the usual set theory. Some of the consistent fragments remain alien to

We then discuss positive set theories. Here we focus on two systems. The most mathematically fluent system of positive set theory is the system

We discuss a pair of set theories motivated by the surprising application of model theory to analysis subsumed under the term “nonstandard analysis”. One of these, Nelson’s

Finally, we discuss a curiosity, which honestly exhibits a characteristic which has been ascribed by philosophers to all systems of set theory (we believe incorrectly). The double extension set theory of Andrzej Kisielewicz is a fascinating and bizarre system of set theory which is frankly an ad hoc solution to the paradoxes. Appropriately, we do not yet know whether this system is consistent. It is at least as strong as

A final curiosity is the theory obtained by adjoing the existence of an elementary embedding j : V → V to Zermelo set theory with the Rank Axiom, which has been studied by Paul Corazza. The existence of such an embedding is inconsistent with

Zermelo set theory was not in our view motivated by an ad hoc attempt to resolve the paradoxes. Zermelo appears to have catalogued constructions in set theory which had actual applications in mathematical reasoning and which were required to implement his proof that the axiom of choice implies the well-ordering theorem. The motivation of simple type theory seems to us to be very similar, though its history is enormously complicated. New Foundations does seem to have started out as an

A class of set theories which we do

Bibliography

Forti-Honsell, Forti-Hinnion, Forster: axioms of set theory.

%Set Theories

%. Randall Holmes and Thomas Forster

basics: type theory and the original theory of Zermelo

In this section we describe two systems which are not “alternative”: they are standard theories. These are simple type theory (in a streamlined form—not the complex theory of {}—and the original set theory of Zermelo, which differs in some ways from its final modern development.

type theory

The simple type theory

The intuition is that type 0 is inhabited by some objects of an unspecified nature called “individuals”, type 1 is inhabited by sets of individuals, type 2 is inhabited by sets of sets of individuals, and so forth; in general type n + 1 is inhabited by sets of type n objects.

There are two axiom schemes which make up the core of the theory and two further axiom schemes which are usually assumed. We use variables without type superscripts: each axiom is asserted for every assignment of types to the variables appearing in it which makes sense.

(∀AB.A=B↔︎(∀x.x∈A↔︎x∈B)). Objects of any positive type are equal iff they have the same elements.

For any formula ϕ in which A does not appear free, (∃A.(∀x.x∈A↔︎ϕ)). Any condition on objects of a given type defines a set of the next higher type.

For any formula ϕ in which Aⁿ + 1 does not appear free, $\x^n\mid \phi\^n+1$ is defined as the unique Aⁿ + 1 such that (∀xⁿ.xⁿ∈Aⁿ+1↔︎ϕ) (which exists by Comprehension and is unique by Extensionality) (we do reserve the right to use this notation without explicit superscripts as we did in the statement of the axiom (schemes)).

We define Vⁿ + 1 as $\x^n \mid x^n=x^n\^n+1$ and ∅ⁿ + 1 as $\x^n \mid x^n\neq x^n\^n+1$. We define $\x^n,y^n\^n+1$ as $^n z^n=xn z^n=yn$. We reserve the right to use these notations without type indices where types can be deduced from context or should be be taken to be completely general.

Ordered pairs can be defined using the usual Kuratowski definition $(x,y) = \{x\,\x,y\\}$, and relations and functions can then be defined as usual. It is unfortunate that in the notation x R y the type of R is 3 higher than that of x or y, and similarly the type of f is three higher than that of x in the notation f(x); there is a way to avoid this (by developing an ordered pair which is of the same type as its projections) but that development would take us a bit far afield.

We define 0 as $\emptyset$: note that we can assign any type ≥ 2 to this object, and in fact we define 0 in each type at or above 2. For each set A, define A + 1 as $\a \cup {x\ \mid a \in A \wedge x \not\in a\}$. A + 1 is the set of all objects obtained by adding a single new element to an element of A. Now 0 + 1 is the set of all one-element sets, which we call 1, and 1 + 1 is the set of all two-element sets, which we call 2, and so forth. We define ℕ, the set of natural numbers, as the intersection of all sets which contain 0 and are closed under the extended “successor” operation. Notice that a version of each natural number is defined in each type with index ≥ 2, and a set ℕ is defined in each type with index ≥ 3. The elements of the natural numbers are the finite sets: we define the set 𝔽 of finite sets as ⋃ℕ.

We can now assert the remaining axioms usually assumed in

Vⁿ + 1 ∉ 𝔽ⁿ + 2. The universe (of objects of a given type) is not a finite set.

Any pairwise disjoint collection of sets has a choice set.

This theory figures in our discussion here primarily as the basis for the development of New Foundations and related theories, but there are other interesting relationships between

This theory has been described carelessly as the simple type theory of Russell, but this is historically inaccurate. Russell had something like this in mind in the Appendix to

original system of Zermelo

The axioms of Zermelo’s original theory of 1908 are presented. The content is presented (this is not a translation). Familiarity with common set theoretical notation is presumed.

If every element of a set A is also an element of B and

∅ exists; for any object x, $$ exists; for any objects x and y, $,y$ exists.

If P(x) is a proposition with a definite truth-value for each x ∈ M, $MP(x)$ exists.

For every set A, the power set $\cal P(A)$, the set of all subsets of A, exists.

For any set A, the union ⋃A, the set of all elements of elements of A exists.

Any pairwise disjoint collection of nonempty sets has a choice set.

There is a set Z such that ∅ ∈ Z and for each x ∈ Z, $\x\ \in Z$.

The axiom of extensionality appears to be phrased in a way which allows for non-set elements of the domain of discourse (objects with no elements distinct from one another and from the empty set). Whether modification accepted here (and your comment deleted; this is what I will do with marginal comments when I accept the change you make, and later in this document I’ll do it without comment) this was intended by Zermelo or not, this alternative possibility is important.

The axiom of elementary sets is usually replaced with the axiom of pairing, which asserts the existence of $,y$ for any objects x and y. The existence of ∅ follows from Separation and the existence of any specific set (such as the one given in Infinity). It is a more modern observation that the existence of $ = ,x$ is a special case of pairing.

Zermelo did not have the modern formulation of the conditions in the axiom of separation in terms of formulas of the first-order language of set theory, and apparently when he saw this formulation he did not like it: he believed that it was too restrictive. The true formulation of Zermelo’s intentions may be second-order.

The axiom of choice takes a modern form. It is interesting that Zermelo proves his theorem that the well-ordering principle follows from choice in the 1908 paper, without having the ability to express ordered pairs as sets (which was present in this theory of course, but first recognized by Norbert Wiener in 1914). want to go look at this proof, though comments about it are unlikely to appear in this venue

The axiom of infinity differs from its usual modern form which asserts the existence of a set which contains ∅ and is closed under the von Neumann successor operation $x x $. A different definition of the natural numbers is the reason: Zermelo defines the natural numbers as the iterated singletons of the empty set. It is an interesting fact that though the mathematical effect of either axiom of infinity is the same, these axioms are not equivalent in the presence of the other axioms of Zermelo set theory: neither one implies the other. They have the same consistency strength: each theory can interpret the other. The two axioms of infinity are equivalent in the presence of the very powerful Axiom of Replacement; they are also equivalent in the presence of a weaker axiom which asserts that each object belongs to some rank of the cumulative hierarchy.traffic on FOM about tis: see mathias JSL recently too. I shall dig this stuff up [do tell! —MRH]

The axiom of foundation is absent from Zermelo’s original theory though it often appears in modern lists of axioms for Zermelo set theory.

For any set x, there is y ∈ x such that y ∩ x = ∅.

This axiom asserts that the membership relation restricted to any set is a well-founded relation. It rules out such things as self-membered sets. This is a consequence of the view that the world of set theory is constructed by the iterated application of the power set operation to an initial collection (the empty set or perhaps a set of atoms) through stages indexed by the ordinals. We will see below that variations of

relationship between these systems. Mac Lane set theory

The simple theory of types and Zermelo set theory appear superficially to live at the same level of strength. Both of them can talk about an infinite set (type 0 for

The source of strength in Zermelo set theory is the ability to quantify over the entire universe in instances of Separation. The variant of Zermelo set theory in which Separation is restricted to Δ₀ formulas (those in which every quantifier is bounded in a set) [with the von Neumann form of Infinity and the Axiom of Foundation] is known as Mac Lane set theory (because it was advocated as a foundational system by Saunders Mac Lane), and Mac Lane set theory is demonstrably mutually interpretable with

Adrian Mathias has written a beautiful survey of relations between Mac Lane set theory, Zermelo set theory, and some other systems, found in , .

Lane or Zermelo set theory as an alternative set theory

As witnessed by the program of Mac Lane mentioned above, Zermelo set theory or variants of Zermelo set theory have been pressed into service themselves as alternative set theories, presumably by workers nervous about the high consistency strength of

One is then faced with the problem that the implementations of cardinal and ordinal numbers traditional in

An elegant solution is obtained by adding the Axiom of Foundation (or restricting our attention to well-founded sets), and further stipulating that every set belongs to a rank of the cumulative hierarchy. It is then possible to define the cardinality of a set A as the set of all sets B which are equinumerous with A and of the lowest rank which contains such sets, and define the order type of a well-ordering ≤ as the set of all well-orderings similar to ≤ and of the lowest rank which contains such well-orderings.

It remains to explain how to stipulate that every set belongs to a rank of the cumulative hierarchy.

A set H is a

For every set x there is a rank r such that x ∈ r.

For any subhierarchies H₁ and H₂, either H₁ ⊆ H₂ or H₂ ⊆ H₁.

omitted

Ranks are well-ordered by inclusion.

For any set x, the rank of x is the minimal rank which contains x as a subset.

For any set A, $|A| r B A$ where r is the minimal rank which contains a set equinumerous with A. A

For any well-ordering ≤, $

For any ordinal α, V_α is the rank (if there one) such that the order type of the inclusion order on the ranks properly included in α is α.

It is further interesting to observe that the following axioms suffice to present this extension of Zermelo (or Mac Lane) set theory.

Sets with the same elements are the same.

For any set A and (Δ₀) formula ϕ in which B is not free, (∃B.x∈B↔︎x∈A∧ϕ).

For any set A, there is a set $\cal P(A)$ whose members are exactly the subsets of A.

Every set belongs to some rank.

Any pairwise disjoint collection of sets has a choice set.

The Axioms of Pairing and Union in the original Zermelo axiom set turn out to be redundant. If x and y belong to ranks r_x and r_y, $,y = r_x r_y z=x z=y$, where r_x ∪ r_y is a set because it is the larger of the two ranks. If x belongs to the rank r_x, $\bigcup x = {y \in r_x \mid (\exists z.y \in z \wedge z \in x)\}$. The Axiom of Foundation follows quite directly from the Axiom of Rank as well.

A book-length development of set theory in this style is . The axiom of rank adds no essential strength to Zermelo or Mac Lane set theory: see .

with classes

considerations

In any version of set theory, we find ourselves wanting to talk about collections which are not sets as if they were sets. It is easy to see that in a certain sense such talk is harmless: if (M,E) is a model of set theory, we could add $\cal P(M)$ (the true power set of M) as a new domain, and add the membership relation of elements of M in elements of $\cal P(M)$ to obtain a theory with signature $(M,,\cal P(M),E,\in\cap(M \times \cal P(M)))$, then finally identify each preimage of an element of M under E with that element of M and collapse E and $\in\cap(M \times \cal P(M)$ into a single relation appropriately. Of course this requires some strength in the metatheory, but not very much.

In such an extended theory, the domain of the original theory is definable as the collection of elements. In the family of theories with classes that we give first, general objects are called {} and objects which are elements of classes are called {}.

In a theory of this kind we call general objects classes. The primitive relations of the theory are equality and membership as usual.

We give the following

$\tt set(x) \equiv (\exists Y.x\in Y)$: a

We use upper case letters for general variables and lower case letters for variables restricted to the sets.

Any theory of this kind has two characteristic axioms as preamble.

(∀AB.A=B↔︎(∀x.x∈A↔︎x∈B))

Where ϕ is a formula which all quantifiers are restricted to the domain of sets in which A does not appear free, (∃A.(∀x.x∈A↔︎ϕ)). Note the use of the variable case convention here: if we were to assume that variables ranged over all classes, this would be $(\exists A.(\forall x.x\in A \leftrightarrow \tt set(x) \wedge \phi))$.

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

The differences between predicative and impredicative class comprehension will be brought out the in the next subsection. The motivating discussion above suggests that any set theory can be extended to a theory of classes with extensionality and full class comprehension. Thus impredicative class comprehension as such adds no additional strength to a set theory, but it does add considerable expressive power.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

Notice that if x and y are sets, the existence of $,y$, ⋃x, $\cal P(x)$ (understood as the collection of sub{} of x) follows from Class Comprehension: the additional axioms corresponding to the axioms of pairing, union, and power set assert that these classes are sets.

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The Axiom of Infinity implies that there are at least three distinct sets (say the von Neumann numerals 0, 1, 2) so there cannot be a bijection between a class $,y$ and the universe, so $,y$ must be a set so Pairing holds. Since Pairing holds, ordered pairs of sets are sets, and so every logical relation on sets is realized by a class of ordered pairs.

The Axiom of Separation follows from the Axiom of Power Set, because of the precise way we phrased it. $y $ exists as a class by class comprehension and is obviously a subclass of y, so an element of $\cal P(y)$, so a set.

The Axiom of Choice (in a strong form) follows from Limitation of Size (surprise!). The class of von Neumann ordinals is not a set, so there is a class bijection from the class of von Neumann ordinals to the class of all sets, which determines a well-ordering of the universe in the obvious way. From any element of a collection of pairwise disjoint sets, select the least element with respect to this global well-ordering: collect these least elements to build a choice set.

The Axiom of Replacement follows from Limitation of Size. Suppose a is a set and (∀x∈a.(∃!y.ϕ)). The class $B = (x a.)$ exists as a class by Class Comprehension. Note that $B = F"a$ where $F = (x,y) $. Define F⁻1(b) as the least element of the preimage of $ under F with respect to a fixed global well-ordering: notice that F⁻1 is an injection from B into a. modified the proof here to use Separation (as a consequence of Power Set) and removed the remarks about deriving Separation from Limitation of Size; I think it can be done but it is rather involved and beside the point being made here).$\F^{-1(b)\mid b \in B\}$ is a set because it is a subclass of a: if there were a class bijection from V to B, there would be a class bijection from V to this set, which contradicts Limitation of Size. Thus B is a set and Replacement holds.

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

To handle classes in which the top level logical operator in the defining formula is propositional, note that ${x } = $ and ${x } =V-$, and every propositional operator can be defined in terms of conjunction and negation. So we stipulate that if A and B are classes, A ∩ B and V − A are classes.

We stipulate that for any sets x and y, $,y$ is a set and so $(x,y) = \{x\,\x,y\\}$ is a set. We define (a₁,a₂,…,a_n) as (a₁,(a₂,…,a_n)). We stipulate that (x₁) = x₁: a 1-tuple is its sole item.

We define π₁ as $\[](http://orpeth.com/.html)$. Objects of any positive type are equal iff they have the same elements.

For any formula ϕ in which A does not appear free, (∃A.(∀x.x∈A↔︎ϕ)). Any condition on objects of a given type defines a set of the next higher type.

We can now assert the remaining axioms usually assumed in

Vⁿ + 1 ∉ 𝔽ⁿ + 2. The universe (of objects of a given type) is not a finite set.

Any pairwise disjoint collection of sets has a choice set.

This theory figures in our discussion here primarily as the basis for the development of New Foundations and related theories, but there are other interesting relationships between

This theory has been described carelessly as the simple type theory of Russell, but this is historically inaccurate. Russell had something like this in mind in the Appendix to

original system of Zermelo

The axioms of Zermelo’s original theory of 1908 are presented. The content is presented (this is not a translation). Familiarity with common set theoretical notation is presumed.

If every element of a set A is also an element of B and

∅ exists; for any object x, $$ exists; for any objects x and y, $,y$ exists.

If P(x) is a proposition with a definite truth-value for each x ∈ M, $MP(x)$ exists.

For every set A, the power set $\cal P(A)$, the set of all subsets of A, exists.

For any set A, the union ⋃A, the set of all elements of elements of A exists.

Any pairwise disjoint collection of nonempty sets has a choice set.

There is a set Z such that ∅ ∈ Z and for each x ∈ Z, $\x\ \in Z$.

The axiom of foundation is absent from Zermelo’s original theory though it often appears in modern lists of axioms for Zermelo set theory.

For any set x, there is y ∈ x such that y ∩ x = ∅.

relationship between these systems. Mac Lane set theory

The simple theory of types and Zermelo set theory appear superficially to live at the same level of strength. Both of them can talk about an infinite set (type 0 for

Adrian Mathias has written a beautiful survey of relations between Mac Lane set theory, Zermelo set theory, and some other systems, found in , .

Lane or Zermelo set theory as an alternative set theory

One is then faced with the problem that the implementations of cardinal and ordinal numbers traditional in

It remains to explain how to stipulate that every set belongs to a rank of the cumulative hierarchy.

A set H is a

For every set x there is a rank r such that x ∈ r.

For any subhierarchies H₁ and H₂, either H₁ ⊆ H₂ or H₂ ⊆ H₁.

omitted

Ranks are well-ordered by inclusion.

For any set x, the rank of x is the minimal rank which contains x as a subset.

For any set A, $|A| r B A$ where r is the minimal rank which contains a set equinumerous with A. A

For any well-ordering ≤, $

For any ordinal α, V_α is the rank (if there one) such that the order type of the inclusion order on the ranks properly included in α is α.

It is further interesting to observe that the following axioms suffice to present this extension of Zermelo (or Mac Lane) set theory.

Sets with the same elements are the same.

For any set A and (Δ₀) formula ϕ in which B is not free, (∃B.x∈B↔︎x∈A∧ϕ).

For any set A, there is a set $\cal P(A)$ whose members are exactly the subsets of A.

Every set belongs to some rank.

Any pairwise disjoint collection of sets has a choice set.

A book-length development of set theory in this style is . The axiom of rank adds no essential strength to Zermelo or Mac Lane set theory: see .

with classes

considerations

In a theory of this kind we call general objects classes. The primitive relations of the theory are equality and membership as usual.

We give the following

$\tt set(x) \equiv (\exists Y.x\in Y)$: a

We use upper case letters for general variables and lower case letters for variables restricted to the sets.

Any theory of this kind has two characteristic axioms as preamble.

(∀AB.A=B↔︎(∀x.x∈A↔︎x∈B))

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $\[](http://orpeth.com/.html)$. Any condition on objects of a given type defines a set of the next higher type.

We can now assert the remaining axioms usually assumed in

Vⁿ + 1 ∉ 𝔽ⁿ + 2. The universe (of objects of a given type) is not a finite set.

Any pairwise disjoint collection of sets has a choice set.

This theory figures in our discussion here primarily as the basis for the development of New Foundations and related theories, but there are other interesting relationships between

This theory has been described carelessly as the simple type theory of Russell, but this is historically inaccurate. Russell had something like this in mind in the Appendix to

original system of Zermelo

The axioms of Zermelo’s original theory of 1908 are presented. The content is presented (this is not a translation). Familiarity with common set theoretical notation is presumed.

If every element of a set A is also an element of B and

∅ exists; for any object x, $$ exists; for any objects x and y, $,y$ exists.

If P(x) is a proposition with a definite truth-value for each x ∈ M, $MP(x)$ exists.

For every set A, the power set $\cal P(A)$, the set of all subsets of A, exists.

For any set A, the union ⋃A, the set of all elements of elements of A exists.

Any pairwise disjoint collection of nonempty sets has a choice set.

There is a set Z such that ∅ ∈ Z and for each x ∈ Z, $\x\ \in Z$.

The axiom of foundation is absent from Zermelo’s original theory though it often appears in modern lists of axioms for Zermelo set theory.

For any set x, there is y ∈ x such that y ∩ x = ∅.

relationship between these systems. Mac Lane set theory

The simple theory of types and Zermelo set theory appear superficially to live at the same level of strength. Both of them can talk about an infinite set (type 0 for

Adrian Mathias has written a beautiful survey of relations between Mac Lane set theory, Zermelo set theory, and some other systems, found in , .

Lane or Zermelo set theory as an alternative set theory

One is then faced with the problem that the implementations of cardinal and ordinal numbers traditional in

It remains to explain how to stipulate that every set belongs to a rank of the cumulative hierarchy.

A set H is a

For every set x there is a rank r such that x ∈ r.

For any subhierarchies H₁ and H₂, either H₁ ⊆ H₂ or H₂ ⊆ H₁.

omitted

Ranks are well-ordered by inclusion.

For any set x, the rank of x is the minimal rank which contains x as a subset.

For any set A, $|A| r B A$ where r is the minimal rank which contains a set equinumerous with A. A

For any well-ordering ≤, $

For any ordinal α, V_α is the rank (if there one) such that the order type of the inclusion order on the ranks properly included in α is α.

It is further interesting to observe that the following axioms suffice to present this extension of Zermelo (or Mac Lane) set theory.

Sets with the same elements are the same.

For any set A and (Δ₀) formula ϕ in which B is not free, (∃B.x∈B↔︎x∈A∧ϕ).

For any set A, there is a set $\cal P(A)$ whose members are exactly the subsets of A.

Every set belongs to some rank.

Any pairwise disjoint collection of sets has a choice set.

A book-length development of set theory in this style is . The axiom of rank adds no essential strength to Zermelo or Mac Lane set theory: see .

with classes

considerations

In a theory of this kind we call general objects classes. The primitive relations of the theory are equality and membership as usual.

We give the following

$\tt set(x) \equiv (\exists Y.x\in Y)$: a

We use upper case letters for general variables and lower case letters for variables restricted to the sets.

Any theory of this kind has two characteristic axioms as preamble.

(∀AB.A=B↔︎(∀x.x∈A↔︎x∈B))

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $[](http://orpeth.com/.html).

We can now assert the remaining axioms usually assumed in

Vⁿ + 1 ∉ 𝔽ⁿ + 2. The universe (of objects of a given type) is not a finite set.

Any pairwise disjoint collection of sets has a choice set.

This theory figures in our discussion here primarily as the basis for the development of New Foundations and related theories, but there are other interesting relationships between

This theory has been described carelessly as the simple type theory of Russell, but this is historically inaccurate. Russell had something like this in mind in the Appendix to

original system of Zermelo

The axioms of Zermelo’s original theory of 1908 are presented. The content is presented (this is not a translation). Familiarity with common set theoretical notation is presumed.

If every element of a set A is also an element of B and

∅ exists; for any object x, $$ exists; for any objects x and y, $,y$ exists.

If P(x) is a proposition with a definite truth-value for each x ∈ M, $MP(x)$ exists.

For every set A, the power set $\cal P(A)$, the set of all subsets of A, exists.

For any set A, the union ⋃A, the set of all elements of elements of A exists.

Any pairwise disjoint collection of nonempty sets has a choice set.

There is a set Z such that ∅ ∈ Z and for each x ∈ Z, $\x\ \in Z$.

The axiom of foundation is absent from Zermelo’s original theory though it often appears in modern lists of axioms for Zermelo set theory.

For any set x, there is y ∈ x such that y ∩ x = ∅.

relationship between these systems. Mac Lane set theory

The simple theory of types and Zermelo set theory appear superficially to live at the same level of strength. Both of them can talk about an infinite set (type 0 for

Adrian Mathias has written a beautiful survey of relations between Mac Lane set theory, Zermelo set theory, and some other systems, found in , .

Lane or Zermelo set theory as an alternative set theory

One is then faced with the problem that the implementations of cardinal and ordinal numbers traditional in

It remains to explain how to stipulate that every set belongs to a rank of the cumulative hierarchy.

A set H is a

For every set x there is a rank r such that x ∈ r.

For any subhierarchies H₁ and H₂, either H₁ ⊆ H₂ or H₂ ⊆ H₁.

omitted

Ranks are well-ordered by inclusion.

For any set x, the rank of x is the minimal rank which contains x as a subset.

For any set A, $|A| r B A$ where r is the minimal rank which contains a set equinumerous with A. A

For any well-ordering ≤, $

For any ordinal α, V_α is the rank (if there one) such that the order type of the inclusion order on the ranks properly included in α is α.

It is further interesting to observe that the following axioms suffice to present this extension of Zermelo (or Mac Lane) set theory.

Sets with the same elements are the same.

For any set A and (Δ₀) formula ϕ in which B is not free, (∃B.x∈B↔︎x∈A∧ϕ).

For any set A, there is a set $\cal P(A)$ whose members are exactly the subsets of A.

Every set belongs to some rank.

Any pairwise disjoint collection of sets has a choice set.

A book-length development of set theory in this style is . The axiom of rank adds no essential strength to Zermelo or Mac Lane set theory: see .

with classes

considerations

In a theory of this kind we call general objects classes. The primitive relations of the theory are equality and membership as usual.

We give the following

$\tt set(x) \equiv (\exists Y.x\in Y)$: a

We use upper case letters for general variables and lower case letters for variables restricted to the sets.

Any theory of this kind has two characteristic axioms as preamble.

(∀AB.A=B↔︎(∀x.x∈A↔︎x∈B))

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $\[](http://orpeth.com/.html))$, then finally identify each preimage of an element of M under E with that element of M and collapse E and $\in\cap(M \times \cal P(M)$ into a single relation appropriately. Of course this requires some strength in the metatheory, but not very much.

In a theory of this kind we call general objects classes. The primitive relations of the theory are equality and membership as usual.

We give the following

$\tt set(x) \equiv (\exists Y.x\in Y)$: a

We use upper case letters for general variables and lower case letters for variables restricted to the sets.

Any theory of this kind has two characteristic axioms as preamble.

(∀AB.A=B↔︎(∀x.x∈A↔︎x∈B))

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $\[](http://orpeth.com/.html)$

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $\[](http://orpeth.com/.html)$. Note the use of the variable case convention here: if we were to assume that variables ranged over all classes, this would be $(\exists A.(\forall x.x\in A \leftrightarrow \tt set(x) \wedge \phi))$.

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $\[](http://orpeth.com/.html)$.

The object A such that (∀x.x∈A↔︎ϕ) (where A is not free in ϕ) is denoted by $$.

Neumann-G"odel-Bernays and Morse-Kelley set theory

Two theories with sets and classes are given here. The theory of sets in each extends

as above

for any class x, there is a class which contains as its elements all the sub

for any class x, ⋃x is a set.

There is a set which contains ∅ and contains $x $ if it contains x, for any set x.

A class C is not a set iff there is a class bijection from the class V of all sets to C.

For any set x, there is an element y of x such that x ∩ y = ∅.

Many of the axioms of

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $\[](http://orpeth.com/.html)$. The class $B = (x a.)$ exists as a class by Class Comprehension. Note that $B = F"a$ where $F = (x,y) $. Define F⁻1(b) as the least element of the preimage of $ under F with respect to a fixed global well-ordering: notice that F⁻1 is an injection from B into a. modified the proof here to use Separation (as a consequence of Power Set) and removed the remarks about deriving Separation from Limitation of Size; I think it can be done but it is rather involved and beside the point being made here).$\F^{-1(b)\mid b \in B\}$ is a set because it is a subclass of a: if there were a class bijection from V to B, there would be a class bijection from V to this set, which contradicts Limitation of Size. Thus B is a set and Replacement holds.

The role of Limitation of Size in encapsulating the powerful axioms of Replacement and Choice is one of the striking features of

We present a reduction of Predicative Class Comprehension to finitely many axioms. This proceeds by induction on the structure of formulas.

We define π₁ as $\[](http://orpeth.com/.html)$. We stipulate that (x₁) = x₁: a 1-tuple is its sole item.

We define π₁ as $((x,y),x)x=x$ and π₂ as $((x,y),y)x=x$. We define A|B for A and B classes as $(x,y) (z.(x,z) A (z,y) B)$. Note that if f and g are functions in the usual sense, f ∘ g = g|f. We stipulate that π₁ is a class, π₂ is a class, and if A and B are classes, so is A|B. Note that each of the sets $_i^n = ((a_1,,a_i,,a_n),a_i)a_1=a_1$, the function sending an n-tuple to its ith element, is a set constructible from the projection functions and relative product, except for $_1^1 = (x,x)x=x$, which we stipulate is a set.

For any set R we define $\tt dom(R)$ as ${x \mid (\exists y.(x,y)\in R)\}$ and R⁻1 as $(y,x)(x,y)R$.

Define R ⊗ S as (R|π₁⁻1) ∩ (S|π₂⁻1). Define R₁ ⊗ R₂ ⊗ … ⊗ R_n as R₁ ⊗ (R₂⊗…⊗R_n) (same grouping as for n-tuples).

Our strategy is first to fix a finite sequence of variables x₁, …, x_n and show how to define ${(x_1,\ldots,x_n)\mid \phi\}$ for any formula in which no variable other than the x_i’s appears free.

We provide $(x_1,x_2) x_1=x_2$ and $(x_1,x_2) x_1x_2$ as basic classes (actually, we have already provided the former).

If we have already defined $X_= (b_1,b_p)$, where all free variables in ψ appear among the b_i’s, then $\(a_1,\ldots,a_n) \mid \psi(a_{s_1,\ldots,a_s_p)\}$ can be shown to exist using the constructions we have already defined and one additional construction. It is constructible as ${\tt dom}((\pi^s_1_n \otimes \ldots \otimes \pi^s_p_n) \cap V \times X_\psi)$, so we stipulate that for any class X, V × X is a class (and for any classes X and Y, X × Y = (V×X)⁻1 ∩ (V×Y) is thus a class).

If we have defined $X_=(a_1,,a_n) $, we construct $(a_1,,a_n) (a_i.)$ as ${\tt dom}(\pi_n^1\otimes \ldots \otimes \pi_n^i-1 \otimes (V\times V)\otimes \pi_n^i+1\otimes \ldots \otimes \pi_n^n)\cap (V \times X_\phi)$. Of course there are typographical variations of this if i = 1, i = n.

Now we show how to construct an arbitrary $$. Let x, x₂, x₃, …, x_n be all the variables free or bound in ϕ. Construct $(x,x_2,,x_n)$. For each of the x_i’s which is free in ϕ, intersect this class with $(\pi_n^i)^-1"\x_i\$ ($R^-1”A = {x }$.

We present the finite list of axioms which we have just shown to imply predicative class comprehension.

For any classes A and B, $A \cap B = {x \mid x \in A \wedge x \in B\}$ is a class.

For any class A, $V-A = x A$ is a class.

For any sets x and y, $,y = z=x z=y$ is a set.

(x,y) is defined as $\{x\,\x,y\\}$.

$_1 = ((x,y),x)x=x$ is a class. $_2 = ((x,y),y)x=x$ is a class.

For any classes R and S, $R|S = {(x,y) \mid (\exists z.(x,z) \in R \wedge (z,y) \in S)\}$ is a class.

For any class R, $(y,x) (x,y) R$ is a class.

For any class R, $

$(x,x) x=x$ and $(x,y)xy$ are classes.

for any classes A and B, $A B = (a,b) a A b B$ is a class (we only used V × A in the construction above, but one might as well provide the general construction).

Various finite sets of constructions have been given for this purpose. The primitives here are basic operations of the algebras of sets and relations, and the approach is inspired by the work of Givant and Tarski in , in which they show that standard axioms for relation algebra augmented with basic properties of projection operations are sufficient to interpret first-order logic.

One can note here that our elimination of the axiom of pairing from the axioms of

set theory

pocket set theory

with atoms and theories with antifoundation axioms

’s AFA

Nowadays when people use the phrase antifoundation axiom' it is almost always to denote specifically the antifoundation axiom of Forti and Honsell [] and not to refer to any of the other antifoundation axioms that have appeared from time to time. Broadly the phraseantifoundation axiom’ is used to refer to an axiom (scheme) which can be added to the axioms of ZF(C) once the axiom of foundation has been dropped. Thus antifoundation axioms are alternative in the sense of being alternative to the axiom scheme of foundation {}. This is a different sense

of `alternative’ from the way in which NF or GPC are alternative.

Forti-Honsell’s axiom (which is called

’s axiom

Foundations and related systems

comprehension

Foundations

Foundations with urelements

variants of New Foundations

Foundations with three types

set theory

system {}

system of Gilmore

motivated by nonstandard analysis

’s {}

’s alternative set theory

extension set theory

set theory with an elementary embedding

orpeth.com