[FOM] The defence of well-founded set theory

[Stanislav Barov]

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).