FOM: As to a "naivete" of G.Cantor's set theory

[Kanovei]

> From: Alexander Zenkin <alexzen at com2com.ru>
> 
> can anyone to formulate explicite arguments why modern meta-mathematics
> calls  the G.Cantor's set theory by a "naive" theory? What does that
> "naivete" consist in? And what does the "non-naivete" of the modern
> meta-mathematics consist in?
> 
> Thanks in advance,
> 
> Alexander Zenkin

The supposed "naivity" of Cantor's set 
theory amounts to the following observations: 

1) One easily encounters a contradiction (eg, the Russell 
paradox), arguing in accordance with the Cantor principles

2) There is another, non-"naive" approach, called axiomatical 
(eg, the Zermelo-Fraenkel theory ZFC), which is strong enough 
to support all (there are still some doubts about this "all") 
mathematically useful applications of the "naive" theory, 
and accurate enough not to lead to contradictions 
(to be on safe side: not to lead so far). 

V.Kanovei