Skand theory and its applications. (A new look at non-well-founded sets)

A {it skand} is defined as some kind of layered set, given by a sequence langle x_beta:alpha_0lebeta<alpharangle of sets indexed by an ordinal interval [alpha_0,alpha). For every beta one has x_{beta+1}in x_beta, but x_beta may contain other elements besides x_{beta+1}. Unfortunately it is not quite clear how the layering relation and the membership relation work at limit levels between alpha_0 and alpha. The skands are intended to be non-well-founded sets, and the author employs them in a discussion of the various paradoxes that plagued set theory in its infancy: those of Russell, Burali-Forti, Zermelo and others. par I am less than convinced by the author's critique of the resolution of these paradoxes. Russell's paradox gets a lot of attention, so let me remark on that here: working in Neumann-Bernays-Gödel-type set theory (NBG), the author defines R={X:M(X)land Xnotin X}, where M(X) abbreviates "X is a set". Now if one assumes M(R) then one derives (Rnotin R)Leftrightarrow(Rin R) in the usual way; the author argues that this is invalid because M(R) implies that (Rnotin R)Rightarrow(Rin R) has truth-value `false', if one assumes the axiom of regularity. This is an unfortunate and incorrect mix of syntax and semantics: the formal derivation of (Rnotin R)Leftrightarrow(Rin R) does not depend on truth values, only on axioms and rules of derivation (and the contradiction itself does not depend on regularity). par The final part of the paper applies skands in the construction of linear orders, but one readily checks that the author basically works with the lexicographic order of various powers 2^alpha: in the skands used, one has at each x_beta one point, 0 or 1, besides x_{beta+1}, and this makes the translation easy enough.