IV. Consistent Set Theories with a Universal Set [chapter]

Universality in Set Theories  
So the universe of CST may consist only of sets, but not all are wellfounded. Obviously UӇU. U is the complement of ∅, so U is the paradigmatic high set. ∅ is well-founded, and, of course, ∅ӇU. The constant predicate "wf( )" expresses the property of BEING WELL-FOUNDED, defined in the usual sense (using some order relation "<"): (Ӂx)(wf(x) ≡ x=∅ Ӡ (Ӂy)(y ⊆ x Ը (y ≠ ∅ Ը (Ӄz)(zӇy ӟ (Ӂw)(wӇy Ը z < w))))). CST can be phrased as a second order system, quantifying over single-or two-argument open
more » ... o-argument open formula ϕ. One could understand this second order quantification as using classes, but only given a full-blown second order
doi:10.1515/9783110326109.49 fatcat:mmhnrgzcxjhy3i2kk244qzwlde