[FOM] The defence of well-founded set theory
Stanislav Barov
barov at mccme.ru
Thu Sep 29 16:54:24 EDT 2005
Historicaly first attempt against Kantorian set theory was made by Russel
in his Ramified Type Theory where no impredicative definitions are
avaliable. In case it apear to be wery weak (no ways to define the
Dedecind's cuts or free group generated by some set, etc.) another way in
avoidance of reflective paradox formulated in ZF theory as Regularity
Acsiom. This kind of axiom have some forms of different strenght. Mostly
minimal conditions of this expresed in stratification axiom of NF of
W. V. O. Quine. H. Wayl influented intuitionistic criticue formulated his
own version of set theory which was precisly described and improved by
S. Faffermann. As sone in ZF was obvious lack for big
sets Geodel and Bernais make defenitions for notion of
class which isn't set and not contained in other class.
I think mostly obvious failure in fundational motivation and
interpretations of set theory contained Lowenghaim-Scoleem
theorem. Outside of scope of ZF exist not intended models for evry
internal formulation of real analisis. On the other hand in (well-founded)
ZF alvays some cardinals are not reachable. Large classes, such as all
models of some algebras contained in more large classes as "elements", for
example in calass of all algebraic varieties, without contradictions (I
hope).
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