An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott and the philosopher George Boolos.
Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmetic and the other usual number systems, and the theory of relations.
ZU etc.
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 by first-order theories not based on sets. The urelements are not essential in that other mathematical structures can be defined 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 an 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.
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 of levels.
Separation: An axiom schema. For any first-order formula Φ(x) with (bound) variables ranging over the level V, the collection { x∈V : Φ(x)} is also a set. (See Axiom schema of separation.)
Remark: Given the levels established by Creation, this schema establishes the existence of sets and how to form them. It tells us that a level is a set, and all subsets, definable via first-order logic, of levels are also sets. This schema can be seen as an extension of the background logic.
Infinity: There exists at least one limit level. (See Axiom of infinity.)
Remark: Among the sets Separation allows, at least one is infinite. This axiom is primarily mathematical, as there is no need for the actual infinite in other human contexts, the human sensory order being necessarily finite. For mathematical purposes, the axiom "There exists an inductive set" would suffice.
Further existence premises
The following statements, while in the nature of axioms, are not axioms of ZU. Instead, they assert the existence of sets satisfying a stated condition. As such, they are "existence premises," meaning the following. Let X denote any statement below. Any theorem whose proof requires X is then formulated conditionally as "If X holds, then..." Potter defines several systems using existence premises, including the following two: ZfU =df ZU + Ordinals; ZFU =df Separation + Reflection.
Ordinals: For each (infinite) ordinal α, there exists a corresponding level Vα.
Remark: In words, "There exists a level corresponding to each infinite ordinal." Ordinals makes possible the conventional Von Neumann definition of ordinal numbers.
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 Φ with 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 level V, in which all properties Φ holding when the quantified variables range over all levels, also hold when these variables range over V only. Reflection turns Creation, Infinity, Ordinals, and Replacement into theorems (Potter 2004: §13.3).