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Additional info for Analysis and partial differential equations. Dedicated to Mischa Cotlar

Sample text

Obviously, x ⊂ Vα+1 and x ∈ Vα+2 ⊂ V. 6. Theorem. , to the coincidence of the universal class and the von Neumann universe. Suppose that U = V and take a nonempty class X. , rank(x) = α and rank(x) ≤ rank(y) for all y ∈ X. 5 (3), rank(u) < α = rank(x), which contradicts the deﬁnition of α. Hence, x ∩ X = 0. Demonstrate now that the supposition V = U contradicts the axiom of regularity. To this end, apply this axiom to the nonempty class U − V and ﬁnd a set y ∈ U − V satisfying y ∩ (U − V) = 0. 5 (6) we deduce y ∈ V, which contradicts the choice of y.

Chapter 2 Boolean Valued Universes It is the use of various rather unconventional models of set theory that uniﬁes the available nonstandard methods of analysis. In particular, the technique of Boolean valued analysis bases on the properties of a certain cumulative hierarchy V(B) whose every successive ﬂoor comprises all functions with domain on the preceding ﬂoors and range in a complete Boolean algebra B ﬁxed in advance. , the Boolean valued universe V(B) . The idea behind the construction of V(B) is very simple.

Zm } stand for the complete list of variables in the construction of ϕ. Assume further that, for all x, u, v, z1 , . . , zm , the following formula holds: ϕ(x, u, z1 , . . , zm ) ∧ ϕ(x, v, z1 , . . , zm ) → u = v. The formula is predicative since each bound variable of ϕ ranges over sets. 14, there is a class Z such that (∀ x)(∀ u)((x, u) ∈ Z ↔ ϕ(x, u, z1 , . . , zm )). , Un (Z) is provable in NGB. By the axiom of replacement NGB6 , there is a set y satisfying (∀ v)(v ∈ y ↔ (∃ u)((u, v) ∈ Z ∧ u ∈ x)).