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: Ordinals: For each (infinite) ordinal α, there exists a corresponding level Vα.
Reflection: Let Φ denote a first-order formula in which any number of free variables are present.
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 that naive set theory allows.
The iterative conception has gradually become more accepted over time, despite an imperfect understanding of its historical origins.
Scott began with an axiom he declined to name: the atomic formula x∈y implies that y is a set.
Scott's final axiom, the Reflection schema, is identical to the above existence premise bearing the same name, and likewise does duty for ZF's Infinity and Replacement.
Potter (1990, 2004) introduced the idiosyncratic terminology described earlier in this entry, and discarded or replaced all of Scott's axioms except Reflection; the result is ZU.
Thus in ZU, an equivalence class of: 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.
Although Potter (2004) devotes an entire appendix to proper classes, the strength and merits of Scott–Potter set theory relative to the well-known rivals to ZFC that admit proper classes, namely NBG and Morse–Kelley set theory, have yet to be explored.
Unlike nearly all writing on set theory in recent decades, Potter (2004) mentions mereological fusions.
His collections are also synonymous with the "virtual sets" of Willard Quine and Richard Milton Martin: entities arising from the free use of the principle of comprehension that can never be admitted to the universe of discourse.