[FOM] The lure of the infinite

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Thu Feb 16 06:27:15 EST 2006

Quoting Martin Davis <martin at eipye.com>:

> In a recent post, Weaver reminds us that the Russell-Cantor
> antinomies concerning the set-theoretic hierarchy are still
> with us. How can we have a notion of set such that one cannot
> form the set of all the entities that fall under it? Workers 
> in f.o.m. are far from having a good answer to this question. 
> Set theorists may speak of proper classes, but, as one of them 
> asked me not so long ago: am I really comfortable with that? 
> Aren't proper classes really just sets in disguise?

Already Cantor himself made a distinction between infinite sets (of various 
powers) which are dealt with in set theory, and absolute infinite, which is 
too huge to be a set. So there is a sense in which paradoxes never really 
threatened Cantorian set theory, but only Frege's system, which is not the 
same thing. 

This is also a comment on some earlier posting here: 
Although the first explicit occurrence of the interative concept of set is 
in Zermelo 1930, is is arguable that Zermelo had the view implicitly 
already in 1908, and indeed that it can be traced back to Cantor himself 
(or, so Hao Wang argued) - if so, this conception of sets was, again, not 
threatened by the paradoxes. 

Best, Panu

Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy, 
University of Helsinki

Visiting Fellow, 
Institute of Philosophy,
School of Advanced Studies, 
University of London

E-mail: panu.raatikainen at helsinki.fi

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