Language: mono-sorted first order logic with equality ‘‘=" and membership ‘‘∈" , with the following axioms added:
Classes: ∀W⃗ ∃!X:X={y|ϕ∧∃Z(y∈Z)}
In English: every formula ϕ defines a unique class of all
elements satisfying it.
Let ‘‘V" represent the class of all elements.
Let ‘‘ON" be the class of all ordinals in V
Define: R is ranking⟺
R:V→ON∧∀x∈V: R(x)=min {α|∀y∈x:α>R(y)}
Sets: X∈V⟺
∃ ranking R:{R(y)|y∈X}≉V
≈ denotes equinumerousity, i.e. existence of a bijection.
In English: an element is a class for which the class of all ranks of its elements is strictly subnumerous to the universe.
Infinity: ∃x∈V:x≠∅∧∀a∈x∃b∈x:b⊋a
There exists a nonempty element that
is closed under existence of proper
superset.