Non-Well-Founded Set Theory

Abstract

This entry is about two kinds of circularity: objectcircularity, where an object is taken to be part of itself insome sense; and definition circularity, where a collection isdefined in terms of itself. Instances of these two kinds ofcircularity are sometimes problematic, and sometimes not. We areprimarily interested in object circularity in this entry, especiallyinstances which look problematic when one tries to model them in settheory. But we shall also discuss circular definitions., (1) HF is the set of all x such that x isa finite subset of HF., Proof.Let c = {x  ∈  s: x ∉  f(x)}. Suppose towards a contradiction that c ∈  f[s]. Fix a  ∈  s such that c = f(a).Then a ∈ c iff a ∉ f(a) iff a ∉ c., Corollary. For all sets s, ℘s is not a subset of s., Proof. If ℘s ⊆ s, we construct a function f from s onto ℘s: let ff(f) =aif a ∈ s, and otherwise let ff(f) = ∅.So we cannot have ℘ s ⊆s, lest we contradict Cantor's Theorem., 0 < 2 < 4 < 3 < 5 < …, each of whose terms is an element of the previous term., fs(n) = ⟨ hn, tn ⟩ = {{hn}, {{hn, tn}}}; , fs(n+1)) = tn∈{hn, tn}∈fs(n)., fs(0)∋{h0, t0}∋fs(1)∋{h1, t1}∋fs(2)∋…, TheoremAssume AFA. Let G be a graph, let x and y be nodes of G, and let d be the decoration of G.Then the following are equivalent:, p ≡ (G,→, g)   and  q ≡ (G,→, h)., ∀x,y(x=y → ∀z(z ∈ x → z ∈ y))., ∀p,q(p ≡ q → ∀r(r ε p → r ε q))., Proposition. Then every monotone operator F onsets has a least fixed point F* and a greatest fixed point F*. In particular, every polynomial operator on classes has least and greatest fixed points. On classes, the same is true for the larger collection of power polynomial operators. , ⟨ y, z, g⟩ ⋅ ⟨ x, y, f⟩ = ⟨ x, z, g ⋅ f ⟩ , (F + G)(f) (inl x) = Ff(x) , {b : φ[b, a1,…, an]}.