UrelementsFrom Wikipedia, the free encyclopedia

Contents

1 Axiom of extensionality 11.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 In predicate logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 In set theory with ur-elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Kripke–Platek set theory with urelements 32.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Additional assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 New Foundations 53.1 The type theory TST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Quinean set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2.1 Axioms and stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.2 Ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.3 Admissibility of useful large sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Finite axiomatizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Cartesian closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 The consistency problem and related partial results . . . . . . . . . . . . . . . . . . . . . . . . . . 73.6 How NF(U) avoids the set-theoretic paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 The system ML (Mathematical Logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8 Models of NFU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.8.1 Self-sufficiency of mathematical foundations in NFU . . . . . . . . . . . . . . . . . . . . 93.8.2 Facts about the automorphism j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.9 Strong axioms of infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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3.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Scott–Potter set theory 144.1 ZU etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1.3 Further existence premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.1 Scott’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.2 Potter’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Urelement 195.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Urelements in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Quine atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 1

Axiom of extensionality

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom ofextensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory.

1.1 Formal statement

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

∀A ∀B (∀X (X ∈ A ⇐⇒ X ∈ B) ⇒ A = B)

or in words:

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B,then A is equal to B.(It is not really essential that X here be a set — but in ZF, everything is. See Ur-elements below forwhen this is violated.)

The converse, ∀A∀B (A = B ⇒ ∀X (X ∈ A ⇐⇒ X ∈ B)) , of this axiom follows from the substitutionproperty of equality.

1.2 Interpretation

To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that Aand B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and onlyif they have precisely the same members. The essence of this is:

A set is determined uniquely by its members.

The axiom of extensionality can be used with any statement of the form ∃A∀X (X ∈ A ⇐⇒ P (X) ) , where P isany unary predicate that does not mention A, to define a unique setA whose members are precisely the sets satisfyingthe predicate P . We can then introduce a new symbol forA ; it’s in this way that definitions in ordinary mathematicsultimately work when their statements are reduced to purely set-theoretic terms.The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or anequivalent appears in just about any alternative axiomatisation of set theory. However, it may require modificationsfor some purposes, as below.

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2 CHAPTER 1. AXIOM OF EXTENSIONALITY

1.3 In predicate logic without equality

The axiom given above assumes that equality is a primitive symbol in predicate logic. Some treatments of axiomaticset theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality.Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol.Most of the axioms of equality still follow from the definition; the remaining one is

∀A ∀B (∀X (X ∈ A ⇐⇒ X ∈ B) ⇒ ∀Y (A ∈ Y ⇐⇒ B ∈ Y ) )

and it becomes this axiom that is referred to as the axiom of extensionality in this context.

1.4 In set theory with ur-elements

An ur-element is a member of a set that is not itself a set. In the Zermelo–Fraenkel axioms, there are no ur-elements,but they are included in some alternative axiomatisations of set theory. Ur-elements can be treated as a differentlogical type from sets; in this case, B ∈ A makes no sense if A is an ur-element, so the axiom of extensionalitysimply applies only to sets.Alternatively, in untyped logic, we can requireB ∈ A to be false wheneverA is an ur-element. In this case, the usualaxiom of extensionality would then imply that every ur-element is equal to the empty set. To avoid this consequence,we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

∀A ∀B (∃X (X ∈ A) ⇒ [∀Y (Y ∈ A ⇐⇒ Y ∈ B) ⇒ A = B] ).

That is:

Given any set A and any set B, if A is a nonempty set (that is, if there exists a member X of A), then ifA and B have precisely the same members, then they are equal.

Yet another alternative in untyped logic is to defineA itself to be the only element ofA wheneverA is an ur-element.While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustmentinstead.

1.5 See also• Extensionality for a general overview.

1.6 References• Paul Halmos,Naive set theory. Princeton, NJ: D. VanNostrandCompany, 1960. Reprinted by Springer-Verlag,New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN3-540-44085-2.

• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

Chapter 2

Kripke–Platek set theory with urelements

The Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements, basedon the traditional (urelement-free) Kripke-Platek set theory. It is considerably weaker than the (relatively) familiarsystem ZFU. The purpose of allowing urelements is to allow large or high-complexity objects (such as the set ofall reals) to be included in the theory’s transitive models without disrupting the usual well-ordering and recursion-theoretic properties of the constructible universe; KP is so weak that this is hard to do by traditional means.

2.1 Preliminaries

The usual way of stating the axioms presumes a two sorted first order language L∗ with a single binary relationsymbol ∈ . Letters of the sort p, q, r, ... designate urelements, of which there may be none, whereas letters of the sorta, b, c, ... designate sets. The letters x, y, z, ... may denote both sets and urelements.The letters for sets may appear on both sides of ∈ , while those for urelements may only appear on the left, i.e. thefollowing are examples of valid expressions: p ∈ a , b ∈ a .The statement of the axioms also requires reference to a certain collection of formulae called ∆0 -formulae. Thecollection∆0 consists of those formulae that can be built using the constants,∈ ,¬ ,∧ ,∨ , and bounded quantification.That is quantification of the form ∀x ∈ a or ∃x ∈ a where a is given set.

2.2 Axioms

The axioms of KPU are the universal closures of the following formulae:

• Extensionality: ∀x(x ∈ a↔ x ∈ b) → a = b

• Foundation: This is an axiom schema where for every formula ϕ(x) we have ∃aϕ(a) → ∃a (ϕ(a) ∧ ∀x ∈a (¬ϕ(x))) .

• Pairing: ∃a (x ∈ a ∧ y ∈ a)

• Union: ∃a∀c ∈ b∀y ∈ c (y ∈ a)

• Δ0-Separation: This is again an axiom schema, where for every ∆0 -formula ϕ(x) we have the following∃a∀x (x ∈ a↔ x ∈ b ∧ ϕ(x)) .

• ∆0 -Collection: This is also an axiom schema, for every ∆0 -formula ϕ(x, y) we have ∀x ∈ a∃y ϕ(x, y) →∃b∀x ∈ a∃y ∈ b ϕ(x, y) .

• Set Existence: ∃a (a = a)

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4 CHAPTER 2. KRIPKE–PLATEK SET THEORY WITH URELEMENTS

2.2.1 Additional assumptions

Technically these are axioms that describe the partition of objects into sets and urelements.

• ∀p∀a (p = a)

• ∀p∀x (x /∈ p)

2.3 Applications

KPU can be applied to the model theory of infinitary languages. Models of KPU considered as sets inside a maximaluniverse that are transitive as such are called admissible sets.

2.4 See also• Axiomatic set theory

• Admissible set

• Admissible ordinal

• Kripke–Platek set theory

2.5 References• Barwise, Jon (1975), Admissible Sets and Structures, Springer-Verlag, ISBN 3-540-07451-1.

• Gostanian, Richard (1980), “Constructible Models of Subsystems of ZF”, Journal of Symbolic Logic 45: 237–250, doi:10.2307/2273185.

2.6 External links• Logic of Abstract Existence

Chapter 3

New Foundations

In mathematical logic,New Foundations (NF) is an axiomatic set theory, conceived byWillard Van Orman Quine asa simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled “NewFoundations for Mathematical Logic"; hence the name. Much of this entry discusses NFU, an important variant ofNF due to Jensen (1969) and exposited in Holmes (1998).[1] In 1940 and 1951 Quine introduced an extension of NFsometimes called “Mathematical Logic” or “ML”, that included classes as well as sets.New Foundations has a universal set, so it is a non well founded set theory.[2] That is to say, it is a logical theory thatallows infinite descending chains of membership such as …x ∈ x -₁ ∈ …x3 ∈ x2 ∈ x1. It avoids Russell’s paradox byonly allowing stratifiable formulae in the axiom of comprehension. For instance x ∈ y is a stratifiable formula, but x∈ x is not (for details of how this works see below).

3.1 The type theory TST

The primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory oftypes, are equality ( = ) and membership ( ∈ ). TST has a linear hierarchy of types: type 0 consists of individualsotherwise undescribed. For each (meta-) natural number n, type n+1 objects are sets of type n objects; sets of typen have members of type n−1. Objects connected by identity must have the same type. The following two atomicformulas succinctly describe the typing rules: xn = yn and xn ∈ yn+1 . (Quinean set theory seeks to eliminate theneed for such superscripts.)The axioms of TST are:

• Extensionality: sets of the same (positive) type with the same members are equal;

• An axiom schema of comprehension, namely:

If ϕ(xn)is a formula, then the set xn | ϕ(xn)n+1 exists.

In other words, given any formula ϕ(xn) , the formula ∃An+1∀xn[xn ∈ An+1 ↔ ϕ(xn)] is an axiomwhere An+1 represents the set xn | ϕ(xn)n+1 .

This type theory is much less complicated than the one first set out in the Principia Mathematica, which includedtypes for relations whose arguments were not necessarily all of the same type. In 1914, Norbert Wiener showed howto code the ordered pair as a set of sets, making it possible to eliminate relation types in favor of the linear hierarchyof sets described here.

3.2 Quinean set theory

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6 CHAPTER 3. NEW FOUNDATIONS

3.2.1 Axioms and stratification

The well-formed formulas of New Foundations (NF) are the same as the well-formed formulas of TST, but with thetype annotations erased. The axioms of NF are:

• Extensionality: Two objects with the same elements are the same object;

• A comprehension schema: All instances of TST Comprehension but with type indices dropped (and withoutintroducing new identifications between variables).

By convention, NF’s Comprehension schema is stated using the concept of stratified formula and making no directreference to types. A formula ϕ is said to be stratified if there exists a function f from pieces of syntax to the naturalnumbers, such that for any atomic subformula x ∈ y of ϕ we have f(y) = f(x) + 1, while for any atomic subformulax = y of ϕ , we have f(x) = f(y). Comprehension then becomes:

x | ϕ exists for each stratified formula ϕ .

Even the indirect reference to types implicit in the notion of stratification can be eliminated. Theodore Hailperinshowed in 1944 that Comprehension is equivalent to a finite conjunction of its instances,[3] so that NF can be finitelyaxiomatized without any reference to the notion of type.Comprehension may seem to run afoul of problems similar to those in naive set theory, but this is not the case. Forexample, the existence of the impossible Russell class x | x ∈ x is not an axiom of NF, because x ∈ x cannot bestratified.

3.2.2 Ordered pairs

Relations and functions are defined in TST (and in NF and NFU) as sets of ordered pairs in the usual way. Theusual definition of the ordered pair, first proposed by Kuratowski in 1921, has a serious drawback for NF and relatedtheories: the resulting ordered pair necessarily has a type two higher than the type of its arguments (its left and rightprojections). Hence for purposes of determining stratification, a function is three types higher than the members ofits field.If one can define a pair in such a way that its type is the same type as that of its arguments (resulting in a type-levelordered pair), then a relation or function is merely one type higher than the type of the members of its field. HenceNF and related theories usually employ Quine's set-theoretic definition of the ordered pair, which yields a type-levelordered pair. Holmes (1998) takes the ordered pair and its left and right projections as primitive. Fortunately, whetherthe ordered pair is type-level by definition or by assumption (i.e., taken as primitive) usually does not matter.The existence of a type-level ordered pair implies Infinity, and NFU + Infinity interprets NFU + “there is a type levelordered pair” (they are not quite the same theory, but the differences are inessential). Conversely, NFU + Infinity +Choice proves the existence of a type-level ordered pair.

3.2.3 Admissibility of useful large sets

NF (and NFU + Infinity + Choice, described below and known consistent) allow the construction of two kinds of setsthat ZFC and its proper extensions disallow because they are “too large” (some set theories admit these entities underthe heading of proper classes):

• The universal set V. Because x = x is a stratified formula, the universal set V = x | x=x exists by Compre-hension. An immediate consequence is that all sets have complements, and the entire set-theoretic universeunder NF has a Boolean structure.

• Cardinal and ordinal numbers. In NF (and TST), the set of all sets having n elements (the circularity hereis only apparent) exists. Hence Frege's definition of the cardinal numbers works in NF and NFU: a cardinalnumber is an equivalence class of sets under the relation of equinumerosity: the sets A and B are equinumerousif there exists a bijection between them, in which case we write A ∼ B . Likewise, an ordinal number is anequivalence class of well-ordered sets.

3.3. FINITE AXIOMATIZABILITY 7

3.3 Finite axiomatizability

New Foundations can be finitely axiomatized. Two such formulations are given here.

3.4 Cartesian closure

Unfortunately, the category whose objects are the sets of NF and whose morphisms are the functions between thosesets is not cartesian closed;[4] this is a highly desirable property for any set theory to have. Intuitively, it means thatthe functions of NF do not curry as one would normally expect functions to. Furthermore, it means that NF is not atopos.

3.5 The consistency problem and related partial results

The outstanding problem with NF is that it is not known to be relatively consistent to any mainstream mathematicalsystem. NF disproves Choice, and so proves Infinity (Specker, 1953). But it is also known (Jensen, 1969) that theminor(?) modification of allowing urelements (multiple distinct objects lacking members) yields NFU, a theory that isconsistent relative to Peano arithmetic; if Infinity and Choice are added, the resulting theory has the same consistencystrength as type theory with infinity or bounded Zermelo set theory. (NFU corresponds to a type theory TSTU, wheretype 0 has urelements, not just a single empty set.) There are other relatively consistent variants of NF.NFU is, roughly speaking, weaker than NF because in NF, the power set of the universe is the universe itself, whilein NFU, the power set of the universe may be strictly smaller than the universe (the power set of the universe containsonly sets, while the universe may contain urelements). In fact, this is necessarily the case in NFU+"Choice”.Specker has shown that NF is equiconsistent with TST + Amb, where Amb is the axiom scheme of typical ambiguitywhich asserts ϕ ↔ ϕ+ for any formula ϕ , ϕ+ being the formula obtained by raising every type index in ϕ by one.NF is also equiconsistent with the theory TST augmented with a “type shifting automorphism”, an operation whichraises type by one, mapping each type onto the next higher type, and preserves equality and membership relations(and which cannot be used in instances of Comprehension: it is external to the theory). The same results hold forvarious fragments of TST in relation to the corresponding fragments of NF.In the same year (1969) that Jensen proved NFU consistent, Grishin provedNF3 consistent. NF3 is the fragment ofNF with full extensionality (no urelements) and those instances of Comprehension which can be stratified using justthree types. This theory is a very awkward medium for mathematics (although there have been attempts to alleviatethis awkwardness), largely because there is no obvious definition for an ordered pair. Despite this awkwardness,NF3

is very interesting because every infinite model of TST restricted to three types satisfies Amb. Hence for every suchmodel there is a model of NF3 with the same theory. This does not hold for four types: NF4 is the same theory asNF, and we have no idea how to obtain a model of TST with four types in which Amb holds.In 1983, Marcel Crabbé proved consistent a system he called NFI, whose axioms are unrestricted extensionality andthose instances of Comprehension in which no variable is assigned a type higher than that of the set asserted to exist.This is a predicativity restriction, though NFI is not a predicative theory: it admits enough impredicativity to definethe set of natural numbers (defined as the intersection of all inductive sets; note that the inductive sets quantifiedover are of the same type as the set of natural numbers being defined). Crabbé also discussed a subtheory of NFI,in which only parameters (free variables) are allowed to have the type of the set asserted to exist by an instance ofComprehension. He called the result “predicative NF” (NFP); it is, of course, doubtful whether any theory with aself-membered universe is truly predicative. Holmes has shown that NFP has the same consistency strength as thepredicative theory of types of Principia Mathematica without the Axiom of reducibility.

3.6 How NF(U) avoids the set-theoretic paradoxes

NF steers clear of the three well-known paradoxes of set theory. That NFU, a (relatively) consistent theory, alsoavoids the paradoxes increases our confidence in this fact.The Russell paradox: An easy matter; x ∈ x is not a stratified formula, so the existence of x | x ∈ x is not assertedby any instance of Comprehension. Quine presumably constructed NF with this paradox uppermost in mind.

8 CHAPTER 3. NEW FOUNDATIONS

Cantor’s paradox of the largest cardinal number exploits the application of Cantor’s theorem to the universal set.Cantor’s theorem says (given ZFC) that the power set P (A) of any set A is larger than A (there can be no injection(one-to-onemap) fromP (A) intoA ). Now of course there is an injection fromP (V ) intoV , ifV is the universal set!The resolution requires that we observe that |A| < |P (A)|makes no sense in the theory of types: the type of P (A) isone higher than the type of A . The correctly typed version (which is a theorem in the theory of types for essentiallythe same reasons that the original form of Cantor’s theorem works in ZF) is |P1(A)| < |P (A)| , where P1(A) is theset of one-element subsets of A . The specific instance of this theorem that interests us is |P1(V )| < |P (V )| : thereare fewer one-element sets than sets (and so fewer one-element sets than general objects, if we are in NFU). The“obvious” bijection x 7→ x from the universe to the one-element sets is not a set; it is not a set because its definitionis unstratified. Note that in all known models of NFU it is the case that |P1(V )| < |P (V )| << |V | ; Choice allowsone not only to prove that there are urelements but that there are many cardinals between |P (V )| and |V | .We now introduce some useful notions. A set A which satisfies the intuitively appealing |A| = |P1(A)| is said tobe Cantorian: a Cantorian set satisfies the usual form of Cantor’s theorem. A set A which satisfies the furthercondition that (x 7→ x)⌈A , the restriction of the singleton map to A, is a set is not only Cantorian set but stronglyCantorian.The Burali-Forti paradox of the largest ordinal number goes as follows. We define (following naive set theory) theordinals as equivalence classes of well-orderings under similarity. There is an obvious natural well-ordering on theordinals; since it is a well-ordering it belongs to an ordinal Ω . It is straightforward to prove (by transfinite induction)that the order type of the natural order on the ordinals less than a given ordinal α is α itself. But this means that Ωis the order type of the ordinals < Ω and so is strictly less than the order type of all the ordinals — but the latter is,by definition, Ω itself!The solution to the paradox in NF(U) starts with the observation that the order type of the natural order on the ordinalsless than α is of a higher type than α . Hence a type level ordered pair is two, and the usual Kuratowski orderedpair, four, types higher than the type of its arguments. For any order type α , we can define an order type α one typehigher: if W ∈ α , then T (α) is the order type of the order W ι = (x, y) | xWy . The triviality of the Toperation is only a seeming one; it is easy to show that T is a strictly monotone (order preserving) operation on theordinals.We can now restate the lemma on order types in a stratified manner: the order type of the natural order on theordinals < α is T 2(α) or T 4(α) depending on which pair is used (we assume the type level pair hereinafter). Fromthis we deduce that the order type on the ordinals < Ω is T 2(Ω) , from which we deduce T 2(Ω) < Ω . Hence theT operation is not a function; we cannot have a strictly monotone set map from ordinals to ordinals which sends anordinal downward! Since T is monotone, we haveΩ > T 2(Ω) > T 4(Ω) . . . , a “descending sequence” in the ordinalswhich cannot be a set.Some have asserted that this result shows that no model of NF(U) is “standard”, since the ordinals in any model ofNFU are externally not well-ordered. We do not take a position on this, but we note that it is also a theorem of NFUthat any set model of NFU has non-well-ordered “ordinals"; NFU does not conclude that the universe V is a modelof NFU, despite V being a set, because the membership relation is not a set relation.For a further development of mathematics in NFU, with a comparison to the development of the same in ZFC, seeimplementation of mathematics in set theory.

3.7 The system ML (Mathematical Logic)

ML is an extension of NF that includes classes as well as sets. The set theory of the 1940 first edition of Quine'sMathematical Logic married NF to the proper classes of NBG set theory, and included an axiom schema of unre-stricted comprehension for proper classes. However J. Barkley Rosser (1942) proved that the system presented inMathematical Logic was subject to the Burali-Forti paradox. This result does not apply to NF. Hao Wang (1950)showed how to amend Quine’s axioms so as to avoid this problem, and Quine included the resulting axiomatizationin the 1951 second and final edition of Mathematical Logic.Wang proved that if NF is consistent then so is ML, and also showed that ML can prove the consistency of NF.

3.8. MODELS OF NFU 9

3.8 Models of NFU

There is a fairly simple method for producing models of NFU in bulk. Using well-known techniques of model theory,one can construct a nonstandard model of Zermelo set theory (nothing nearly as strong as full ZFC is needed for thebasic technique) on which there is an external automorphism j (not a set of the model) which moves a rank Vα ofthe cumulative hierarchy of sets. We may suppose without loss of generality that j(α) < α . We talk about theautomorphism moving the rank rather than the ordinal because we do not want to assume that every ordinal in themodel is the index of a rank.The domain of the model of NFU will be the nonstandard rank Vα . The membership relation of the model of NFUwill be

• x ∈NFU y ≡def j(x) ∈ y ∧ y ∈ Vj(α)+1.

We now prove that this actually is a model of NFU. Let ϕ be a stratified formula in the language of NFU. Choosean assignment of types to all variables in the formula which witnesses the fact that it is stratified. Choose a naturalnumber N greater than all types assigned to variables by this stratification.Expand the formula ϕ into a formula ϕ1 in the language of the nonstandard model of Zermelo set theory withautomorphism j using the definition of membership in the model of NFU. Application of any power of j to bothsides of an equation or membership statement preserves its truth value because j is an automorphism. Make suchan application to each atomic formula in ϕ1 in such a way that each variable x assigned type i occurs with exactlyN − i applications of j. This is possible thanks to the form of the atomic membership statements derived from NFUmembership statements, and to the formula being stratified. Each quantified sentence (∀x ∈ Vα.ψ(j

N−i(x))) can beconverted to the form (∀x ∈ jN−i(Vα).ψ(x)) (and similarly for existential quantifiers). Carry out this transformationeverywhere and obtain a formula ϕ2 in which j is never applied to a bound variable.Choose any free variable y in ϕ assigned type i. Apply ji−N uniformly to the entire formula to obtain a formulaϕ3 in which y appears without any application of j. Now y ∈ Vα | ϕ3 exists (because j appears applied only tofree variables and constants), belongs to Vα+1 , and contains exactly those y which satisfy the original formula ϕ inthe model of NFU. j(y ∈ Vα | ϕ3) has this extension in the model of NFU (the application of j corrects for thedifferent definition of membership in the model of NFU). This establishes that Stratified Comprehension holds in themodel of NFU.To see that weak Extensionality holds is straightforward: each nonempty element of Vj(α)+1 inherits a unique ex-tension from the nonstandard model, the empty set inherits its usual extension as well, and all other objects areurelements.The basic idea is that the automorphism j codes the “power set” Vα+1 of our “universe” Vα into its externally iso-morphic copy Vj(α)+1 inside our “universe.” The remaining objects not coding subsets of the universe are treated asurelements.If α is a natural number n, we get a model of NFU which claims that the universe is finite (it is externally infinite, ofcourse). If α is infinite and the Choice holds in the nonstandard model of ZFC, we obtain a model of NFU + Infinity+ Choice.

3.8.1 Self-sufficiency of mathematical foundations in NFU

For philosophical reasons, it is important to note that it is not necessary to work in ZFC or any related system tocarry out this proof. A common argument against the use of NFU as a foundation for mathematics is that our reasonsfor relying on it have to do with our intuition that ZFC is correct. We claim that it is sufficient to accept TST (infact TSTU). We outline the approach: take the type theory TSTU (allowing urelements in each positive type) as ourmetatheory and consider the theory of set models of TSTU in TSTU (these models will be sequences of sets Ti (allof the same type in the metatheory) with embeddings of each P (Ti) into P1(Ti+1) coding embeddings of the powerset of Ti into Ti+1 in a type-respecting manner). Given an embedding of T0 into T1 (identifying elements of the base“type” with subsets of the base type), one can define embeddings from each “type” into its successor in a natural way.This can be generalized to transfinite sequences Tα with care.Note that the construction of such sequences of sets is limited by the size of the type in which they are being con-structed; this prevents TSTU from proving its own consistency (TSTU + Infinity can prove the consistency of TSTU;to prove the consistency of TSTU+Infinity one needs a type containing a set of cardinality ℶω , which cannot be

10 CHAPTER 3. NEW FOUNDATIONS

proved to exist in TSTU+Infinity without stronger assumptions). Now the same results of model theory can be usedto build a model of NFU and verify that it is a model of NFU in much the same way, with the Tα 's being used inplace of Vα in the usual construction. The final move is to observe that since NFU is consistent, we can drop the useof absolute types in our metatheory, bootstrapping the metatheory from TSTU to NFU.

3.8.2 Facts about the automorphism j

The automorphism j of a model of this kind is closely related to certain natural operations in NFU. For example, ifW is a well-ordering in the nonstandard model (we suppose here that we use Kuratowski pairs so that the coding offunctions in the two theories will agree to some extent) which is also a well-ordering in NFU (all well-orderings ofNFU are well-orderings in the nonstandard model of Zermelo set theory, but not vice versa, due to the formation ofurelements in the construction of the model), and W has type α in NFU, then j(W) will be a well-ordering of typeT(α) in NFU.In fact, j is coded by a function in the model of NFU. The function in the nonstandard model which sends the singletonof any element of Vj(α) to its sole element, becomes in NFU a function which sends each singleton x, where x isany object in the universe, to j(x). Call this function Endo and let it have the following properties: Endo is an injectionfrom the set of singletons into the set of sets, with the property that Endo( x ) = Endo( y ) | y∈x for each set x.This function can define a type level “membership” relation on the universe, one reproducing the membership relationof the original nonstandard model.

3.9 Strong axioms of infinity

In this section we mainly discuss the effect of adding various “strong axioms of infinity” to our usual base theory,NFU + Infinity + Choice. This base theory, known consistent, has the same strength as TST + Infinity, or Zermeloset theory with Separation restricted to bounded formulas (Mac Lane set theory).One can add to this base theory strong axioms of infinity familiar from the ZFC context, such as “there exists aninaccessible cardinal,” but it is more natural to consider assertions about Cantorian and strongly Cantorian sets. Suchassertions not only bring into being large cardinals of the usual sorts, but strengthen the theory on its own terms.The weakest of the usual strong principles is:

• Rosser’s Axiom of Counting. The set of natural numbers is a strongly Cantorian set.

To see how natural numbers are defined in NFU, see set-theoretic definition of natural numbers. The original formof this axiom given by Rosser was “the set m|1≤m≤n has nmembers”, for each natural number n". This intuitivelyobvious assertion is unstratified: what is provable in NFU is “the set m|1≤m≤n has T 2(n) members” (where the Toperation on cardinals is defined by T (|A|) = |P1(A)| ; this raises the type of a cardinal by one). For any cardinalnumber (including natural numbers) to assert T (|A|) = |A| is equivalent to asserting that the setsA of that cardinalityare Cantorian (by a usual abuse of language, we refer to such cardinals as “Cantorian cardinals”). It is straightforwardto show that the assertion that each natural number is Cantorian is equivalent to the assertion that the set of all naturalnumbers is strongly Cantorian.Counting is consistent with NFU, but increases its consistency strength noticeably; not, as one would expect, in thearea of arithmetic, but in higher set theory. NFU + Infinity proves that each ℶn exists, but not that ℶω exists; NFU+ Counting (easily) proves Infinity, and further proves the existence of ℶℶn for each n, but not the existence of ℶℶω

. (See beth numbers).Counting implies immediately that one does not need to assign types to variables restricted to the set N of naturalnumbers for purposes of stratification; it is a theorem that the power set of a strongly Cantorian set is strongly Can-torian, so it is further not necessary to assign types to variables restricted to any iterated power set of the naturalnumbers, or to such familiar sets as the set of real numbers, the set of functions from reals to reals, and so forth.The set-theoretical strength of Counting is less important in practice than the convenience of not having to annotatevariables known to have natural number values (or related kinds of values) with singleton brackets, or to apply the Toperation in order to get stratified set definitions.Counting implies Infinity; each of the axioms below needs to be adjoined to NFU + Infinity to get the effect of strongvariants of Infinity; Ali Enayat has investigated the strength of some of these axioms in models of NFU + “the universeis finite”.

3.9. STRONG AXIOMS OF INFINITY 11

A model of the kind constructed above satisfies Counting just in case the automorphism j fixes all natural numbers inthe underlying nonstandard model of Zermelo set theory.The next strong axiom we consider is the

• Axiomof stronglyCantorian separation: For any strongly Cantorian setA and any formulaϕ (not necessarilystratified!) the set x∈A|φ exists.

Immediate consequences include Mathematical Induction for unstratified conditions (which is not a consequence ofCounting; many but not all unstratified instances of induction on the natural numbers follow from Counting).This axiom is surprisingly strong. Unpublished work of Robert Solovay shows that the consistency strength of thetheory NFU* = NFU + Counting + Strongly Cantorian Separation is the same as that of Zermelo set theory + Σ2

Replacement.This axiom holds in a model of the kind constructed above (with Choice) if the ordinals which are fixed by j anddominate only ordinals fixed by j in the underlying nonstandard model of Zermelo set theory are standard, and thepower set of any such ordinal in the model is also standard. This condition is sufficient but not necessary.Next is

• Axiom of Cantorian Sets: Every Cantorian set is strongly Cantorian.

This very simple and appealing assertion is extremely strong. Solovay has shown the precise equivalence of theconsistency strength of the theory NFUA = NFU + Infinity + Cantorian Sets with that of ZFC + a schema assertingthe existence of an n-Mahlo cardinal for each concrete natural number n. Ali Enayat has shown that the theory ofCantorian equivalence classes of well-founded extensional relations (which gives a natural picture of an initial segmentof the cumulative hierarchy of ZFC) interprets the extension of ZFC with n-Mahlo cardinals directly. A permutationtechnique can be applied to a model of this theory to give a model in which the hereditarily strongly Cantorian setswith the usual membership relation model the strong extension of ZFC.This axiom holds in a model of the kind constructed above (with Choice) just in case the ordinals fixed by j in theunderlying nonstandard model of ZFC are an initial (proper class) segment of the ordinals of the model.Next consider the

• Axiom of Cantorian Separation: For any Cantorian set A and any formula ϕ (not necessarily stratified!) theset x∈A|φ exists.

This combines the effect of the two preceding axioms and is actually even stronger (precisely how is not known).Unstratified mathematical induction enables proving that there are n-Mahlo cardinals for every n, given CantorianSets, which gives an extension of ZFC that is even stronger than the previous one, which only asserts that there aren-Mahlos for each concrete natural number (leaving open the possibility of nonstandard counterexamples).This axiom will hold in a model of the kind described above if every ordinal fixed by j is standard, and every powerset of an ordinal fixed by j is also standard in the underlying model of ZFC. Again, this condition is sufficient but notnecessary.An ordinal is said to be Cantorian if it is fixed by T, and strongly Cantorian if it dominates only Cantorian ordinals (thisimplies that it is itself Cantorian). In models of the kind constructed above, Cantorian ordinals of NFU correspondto ordinals fixed by j (they are not the same objects because different definitions of ordinal numbers are used in thetwo theories).Equal in strength to Cantorian Sets is the

• Axiom of Large Ordinals: For each non-Cantorian ordinal α , there is a natural number n such that Tn(Ω) <α .

Recall that Ω is the order type of the natural order on all ordinals. This only implies Cantorian Sets if we have Choice(but is at that level of consistency strength in any case). It is remarkable that one can even define Tn(Ω) : this is thenth term sn of any finite sequence of ordinals s of length n such that s0 = Ω , si+1 = T (si) for each appropriatei. This definition is completely unstratified. The uniqueness of Tn(Ω) can be proved (for those n for which it exists)

12 CHAPTER 3. NEW FOUNDATIONS

and a certain amount of common-sense reasoning about this notion can be carried out, enough to show that LargeOrdinals implies Cantorian Sets in the presence of Choice. In spite of the knotty formal statement of this axiom, it isa very natural assumption, amounting to making the action of T on the ordinals as simple as possible.A model of the kind constructed above will satisfy Large Ordinals, if the ordinals moved by j are exactly the ordinalswhich dominate some j−i(α) in the underlying nonstandard model of ZFC.

• Axiom of Small Ordinals: For any formula φ, there is a set A such that the elements of A which are stronglyCantorian ordinals are exactly the strongly Cantorian ordinals such that φ.

Solovay has shown the precise equivalence in consistency strength of NFUB = NFU + Infinity + Cantorian Sets +Small Ordinals with Morse–Kelley set theory plus the assertion that the proper class ordinal (the class of all ordinals)is a weakly compact cardinal. This is very strong indeed! Moreover, NFUB-, which is NFUB with Cantorian Setsomitted, is easily seen to have the same strength as NFUB.Amodel of the kind constructed abovewill satisfy this axiom if every collection of ordinals fixed by j is the intersectionof some set of ordinals with the ordinals fixed by j, in the underlying nonstandard model of ZFC.Even stronger is the theory NFUM = NFU + Infinity + Large Ordinals + Small Ordinals. This is equivalent to Morse–Kelley set theory with a predicate on the classes which is a κ-complete nonprincipal ultrafilter on the proper classordinal κ; in effect, this is Morse–Kelley set theory + “the proper class ordinal is a measurable cardinal"!The technical details here are not the main point, which is that reasonable and natural (in the context of NFU)assertions turn out to be equivalent in power to very strong axioms of infinity in the ZFC context. This fact is relatedto the correlation between the existence of models of NFU, described above and satisfying these axioms, and theexistence of models of ZFC with automorphisms having special properties.

3.10 See also

• Alternative set theory

• Axiomatic set theory

• Implementation of mathematics in set theory

• Positive set theory

• Set-theoretic definition of natural numbers

3.11 Notes

[1] Holmes, Randall, 1998. Elementary Set Theory with a Universal Set. Academia-Bruylant.

[2] Quine’s New Foundations - Stanford Encyclopedia of Philosophy

[3] Hailperin, T. “A set of axioms for logic,” Journal of Symbolic Logic 9, pp. 1-19.

[4] http://www.dpmms.cam.ac.uk/~tf/cartesian-closed.pdf

3.12 References

• Crabbé,Marcel, 1982, On the consistency of an impredicative fragment ofQuine’s NF,The Journal of SymbolicLogic 47: 131-136.

• Forster, T. E. (1992), Set theory with a universal set. Exploring an untyped universe, Oxford Science Pub-lications, Oxford Logic Guides 20, New York: The Clarendon Press, Oxford University Press, ISBN 0-19-853395-0, MR 1166801

3.13. EXTERNAL LINKS 13

• Holmes, M. Randall (1998), Elementary set theory with a universal set (PDF), Cahiers du Centre de Logique10, Louvain-la-Neuve: Université Catholique de Louvain, Département de Philosophie, ISBN 2-87209-488-1,MR 1759289

• Jensen, R. B. (1969), “On the Consistency of a Slight(?) Modification of Quine’s NF”, Synthese 19: 250–63,doi:10.1007/bf00568059, JSTOR 20114640 With discussion by Quine.

• Quine, W. V. (1937), “New Foundations for Mathematical Logic”, The American Mathematical Monthly(Mathematical Association of America) 44 (2): 70–80, doi:10.2307/2300564, JSTOR 2300564

• Quine, Willard Van Orman (1940),Mathematical Logic (first ed.), New York: W. W. Norton & Co., Inc., MR0002508

• Quine, Willard Van Orman (1951), Mathematical logic (Revised ed.), Cambridge, Mass.: Harvard UniversityPress, ISBN 0-674-55451-5, MR 0045661

• Quine, W. V., 1980, “New Foundations for Mathematical Logic” in From a Logical Point of View, 2nd ed.,revised. Harvard Univ. Press: 80-101. The definitive version of where it all began, namely Quine’s 1937 paperin the American Mathematical Monthly.

• Rosser, Barkley (1942), “The Burali-Forti paradox”, J. Symbolic Logic 7: 1–17, MR 0006327

• Wang, Hao (1950), “A formal system of logic”, J. Symbolic Logic 15: 25–32, MR 0034733

3.13 External links• http://math.stanford.edu/~feferman/papers/ess.pdf

• Stanford Encyclopedia of Philosophy:

• Quine’s New Foundations — by Thomas Forster.• Alternative axiomatic set theories — by Randall Holmes.

• Randall Holmes: New Foundations Home Page.

• Randall Holmes: Bibliography of Set Theory with a Universal Set.

• Randall Holmes: Symmetry as a Criterion for Comprehension Motivating Quine’s ‘New Foundations’

Chapter 4

Scott–Potter set theory

An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a col-lection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by themathematician Dana Scott and the philosopher George Boolos.Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic settheory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmeticand the other usual number systems, and the theory of relations.

4.1 ZU etc.

4.1.1 Preliminaries

This section and the next follow Part I of Potter (2004) closely. The background logic is first-order logic with identity.The ontology includes urelements as well as sets, which makes it clear that there can be sets of entities defined byfirst-order theories not based on sets. The urelements are not essential in that other mathematical structures can bedefined as sets, and it is permissible for the set of urelements to be empty.Some terminology peculiar to Potter’s set theory:

• ι is a definite description operator and binds a variable. (In Potter’s notation the iota symbol is inverted.)

• The predicate U holds for all urelements (non-collections).

• ιxΦ(x) exists iff (∃!x)Φ(x). (Potter uses Φ and other upper-case Greek letters to represent formulas.)

• x : Φ(x) is an abbreviation for ιy(not U(y) and (∀x)(x ∈ y ⇔ Φ(x))).

• a is a collection if x : x∈a exists. (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 ∀v∈V(v = acc(V∩v)) 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.

• A set is a subcollection of some level.

• The birthday of set a, denoted V(a), is the lowest level V such that a⊂V.

14

4.1. ZU ETC. 15

4.1.2 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 oflevels.Separation: An axiom schema. For any first-order formula Φ(x) with (bound) variables ranging over the level V, thecollection 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 canbe 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 thereis no need for the actual infinite in other human contexts, the human sensory order being necessarily finite. Formathematical purposes, the axiom “There exists an inductive set" would suffice.

4.1.3 Further existence premises

The following statements, while in the nature of axioms, are not axioms of ZU. Instead, they assert the existence ofsets satisfying a stated condition. As such, they are “existence premises,” meaning the following. Let X denote anystatement below. Any theorem whose proof requiresX is then formulated conditionally as “If X holds, then...” Potterdefines several systems using existence premises, including the following two:

• ZfU = ZU + Ordinals;

• ZFU = 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 conven-tional 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.Reflection: Let Φ denote a first-order formula in which any number of free variables are present. Let Φ(V) denote Φwith these free variables all quantified, with the quantified variables restricted to the level V.Then ∃V[Φ→Φ(V)] is an axiom.Remark: This schema asserts the existence of a “partial” universe, namely the levelV, in which all properties Φ holdingwhen the quantified variables range over all levels, also hold when these variables range over V only. Reflection turnsCreation, Infinity, Ordinals, and Replacement into theorems (Potter 2004: §13.3).Let A and a denote sequences of nonempty sets, each indexed by n.Countable Choice: Given any sequence A, there exists a sequence a such that:

∀n∈ω[a ∈A ].

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].

16 CHAPTER 4. SCOTT–POTTER SET THEORY

4.2 Discussion

The Von Neumann universe implements the “iterative conception of set” by stratifying the universe of sets into aseries of “levels,” with the sets at a given level being the members of the sets making up the next higher level. Hencethe 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 paradoxes of Russell,Burali-Forti, and Cantor. These paradoxes all result from the unrestricted use of the principle of comprehension thatnaive set theory allows. Collections such as “the class of all sets” or “the class of all ordinals” include sets from alllevels of the hierarchy. Given the iterative conception, such collections cannot form sets at any given level of thehierarchy 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 theoryinvolving sets and levels.

4.2.1 Scott’s theory

Scott (1974) did not mention the “iterative conception of set,” instead proposing his theory as a natural outgrowth ofthe simple theory of types. Nevertheless, Scott’s theory can be seen as an axiomatization of the iterative conceptionand 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:

∀x,y∃a[x∈y→y=a].

His axiom of Extensionality and axiom schema of Comprehension (Separation) are strictly analogous to their ZFcounterparts 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 definitionof 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, is identical to the above existence premise bearing the same name, andlikewise does duty for ZF’s Infinity and Replacement. Scott’s system has the same strength as ZF.

4.2.2 Potter’s theory

Potter (1990, 2004) introduced the idiosyncratic terminology described earlier in this entry, and discarded or replacedall of Scott’s axioms except Reflection; the result is ZU. ZU, like ZF, cannot be finitely axiomatized. ZU differs fromZFC in that it:

• Includes no axiom of extensionality because the usual extensionality principle follows from the definition ofcollection and an easy lemma.

• Admits nonwellfounded collections. However Potter (2004) never invokes such collections, and all sets (col-lections which are contained in a level) are wellfounded. No theorem in Potter would be overturned if an axiomstating that all collections are sets were added to ZU.

• Includes no equivalents of Choice or the axiom schema of Replacement.

Hence ZU is closer to the Zermelo set theory of 1908, namely ZFC minus Choice, Replacement, and Foundation.It is stronger than this theory, however, since cardinals and ordinals can be defined, despite the absence of Choice,using Scott’s trick and the existence of levels, and no such definition is possible in Zermelo set theory. Thus in ZU,an equivalence class of:

4.3. SEE ALSO 17

• Equinumerous sets from a common level is a cardinal number;

• Isomorphic well-orderings, also from a common level, is an ordinal number.

Similarly the natural numbers 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 a unary injective operation,successor, whose domain contains a unique element, zero, absent from its range. Because the theory of Dedekindalgebras is categorical (all models are isomorphic), any such algebra can proxy for the natural numbers.Although Potter (2004) devotes an entire appendix to proper classes, the strength and merits of Scott–Potter settheory 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 (Jensen 1967) devised axiomatic set theoryadmitting both urelements and sets that are not well-founded. But the urelements of NFU, unlike those of ZU, play anessential role; they and the resulting restrictions on Extensionality make possible a proof of NFU’s consistency relativeto Peano arithmetic. But nothing is known about the strength of NFU relative to Creation+Separation, NFU+Infinityrelative 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 collectionsare also synonymous with the “virtual sets” of Willard Quine and Richard Milton Martin: entities arising from thefree use of the principle of comprehension that can never be admitted to the universe of discourse.

4.3 See also

• Foundation of mathematics

• Hierarchy (mathematics)

• List of set theory topics

• Philosophy of mathematics

• S (Boolos 1989)

• Von Neumann universe

• Zermelo set theory

• ZFC

4.4 References

• George Boolos, 1971, “The iterative conception of set,” Journal of Philosophy 68: 215–31. Reprinted inBoolos 1999. Logic, Logic, and Logic. Harvard Univ. Press: 13-29.

• --------, 1989, “Iteration Again,” Philosophical Topics 42: 5-21. Reprinted in Boolos 1999. Logic, Logic, andLogic. Harvard Univ. Press: 88-104.

• Potter, Michael, 1990. Sets: An Introduction. Oxford Univ. Press.

• ------, 2004. Set Theory and its Philosophy. Oxford Univ. Press.

• Dana Scott, 1974, “Axiomatizing set theory” in Jech, Thomas, J., ed., Axiomatic Set Theory II, Proceedings ofSymposia in Pure Mathematics 13. American Mathematical Society: 207–14.

18 CHAPTER 4. SCOTT–POTTER SET THEORY

4.5 External links

Reviews of Potter (2004):

• Bays, Timothy, 2005, "Review," Notre Dame Philosophical Reviews.

• Uzquiano, Gabriel, 2005, "Review," Philosophia Mathematica 13: 308-46.

Chapter 5

Urelement

In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is anobject (concrete or abstract) that is not a set, but that may be an element of a set. Urelements are sometimes called“atoms” or “individuals.”

5.1 Theory

There are several different but essentially equivalent ways to treat urelements in a first-order theory.One way is to work in a first-order theory with two sorts, sets and urelements, with a ∈ b only defined when b is a set.In this case, if U is an urelement, it makes no sense to say

X ∈ U,

although

U ∈ X,

is perfectly legitimate.This should not be confused with the empty set where saying

X ∈ ∅

is well-formed (albeit false) because ∅ is a set, whereas U is not.Another way is to work in a one-sorted theory with a unary relation used to distinguish sets and urelements. Asnon-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the emptyset from urelements. Note that in this case, the axiom of extensionality must be formulated to apply only to objectsthat are not urelements.This situation is analogous to the treatments of theories of sets and classes. Indeed, urelements are in some sensedual to proper classes: urelements cannot have members whereas proper classes cannot be members. Put differently,urelements are minimal objects while proper classes are maximal objects by the membership relation (which, ofcourse, is not an order relation, so this analogy is not to be taken literally.)

5.2 Urelements in set theory

The Zermelo set theory of 1908 included urelements. It was soon realized that in the context of this and closely relatedaxiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without

19

20 CHAPTER 5. URELEMENT

urelements. Thus standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements.(For an exception, see Suppes.[1]) Axiomatizations of set theory that do invoke urelements include Kripke–Platek settheory with urelements, and the variant of Von Neumann–Bernays–Gödel set theory described by Mendelson.[2] Intype theory, an object of type 0 can be called an urelement; hence the name “atom.”Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences. In particular,Jensen proved[3] the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative toanything remains an open problem. Moreover, NFU remains relatively consistent when augmented with an axiomof infinity and the axiom of choice. Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem.Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF.Holmes further argues that set theory is more natural with than without urelements, since we may take as urelementsthe objects of any theory or of the physical universe.[4]

5.3 Quine atoms

An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particulartype of set. Quine atoms are sets that only contain themselves, that is, sets that satisfy the formula x = x.[5]

Quine atoms cannot exist in systems of set theory that include the axiom of regularity, but they can exist in non-well-founded set theory. ZF set theory with the axiom of regularity removed is compatible with the existence of Quineatoms, although it does not prove that any non-well-founded sets exist. Aczel’s anti-foundation axiom implies thereis a unique Quine atom. Other non-well-founded theories may admit many distinct Quine atoms; at the opposite endof the spectrum lies Boffa’s axiom of superuniversality, which implies that the distinct Quine atoms form a properclass.[6]

Quine atoms also appear in Quine’s New Foundations, which allows more than one such set to exist.[7]

Quine atoms are the only sets called reflexive sets by Aczel,[6] although other authors, e.g. Jon Barwise and LawrenceMoss use the latter term to denote the larger class of sets with the property x ∈ x.[8]

5.4 References[1] Suppes, Patrick (1972). Axiomatic Set Theory ([Éd. corr. et augm. du texte paru en 1960]. ed.). New York: Dover Publ.

ISBN 0486616304. Retrieved 17 September 2012.

[2] Mendelson, Elliott (1997). Introduction to Mathematical Logic (4th ed.). London: Chapman & Hall. pp. 297–304. ISBN978-0412808302. Retrieved 17 September 2012.

[3] Jensen, Ronald Björn (December 1968). “On the Consistency of a Slight (?) Modification of Quine’s 'New Foundations’".Synthese (Springer) 19 (1/2): 250–264. doi:10.1007/bf00568059. ISSN 0039-7857. Retrieved 17 September 2012.

[4] Holmes, Randall, 1998. Elementary Set Theory with a Universal Set. Academia-Bruylant.

[5] Thomas Forster (2003). Logic, Induction and Sets. Cambridge University Press. p. 199. ISBN 978-0-521-53361-4.

[6] Aczel, Peter (1988), Non-well-founded sets (PDF), CSLI Lecture Notes 14, Stanford University, Center for the Study ofLanguage and Information, p. 57, ISBN 0-937073-22-9, MR 0940014

[7] Barwise, Jon; Moss, Lawrence S. (1996),Vicious circles. On themathematics of non-wellfounded phenomena, CSLI LectureNotes 60, CSLI Publications, p. 306, ISBN 1575860090

[8] Barwise, Jon; Moss, Lawrence S. (1996),Vicious circles. On themathematics of non-wellfounded phenomena, CSLI LectureNotes 60, CSLI Publications, p. 57, ISBN 1575860090

5.5 External links• Weisstein, Eric W., “Urelement”, MathWorld.

5.6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 21

5.6 Text and image sources, contributors, and licenses

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