[FOM] Paradoxes and Platonism

Steven Gubkin steven.gubkin at case.edu
Tue May 20 12:14:27 EDT 2008

If the universe of sets has objective reality, then you cannot just
define whatever new set you want. If you look around the universe of
sets you wil not find the set R; the paradox arises when you try to
introduce this new set into the universe.

Here is a related example. Say we have a library (an objective physical
library). Some of the books are lists of other books and some are not
("Catalog the AMS" is a list of publications, "Animorphs volume 6" is
not). Now write a book R which is a list of all the lists in the library
which do not mention themselves. If you want R  to be part of the
library and be accurate, Russel's paradox stops you. Despite this I have
no trouble believing in the objective reality of libraries. There is no
problem if R is outside of the library.

> But suppose we take the set theoretic universe as an objective and 
> intemporally given reality so that, for instance, it is an 
> objective fact for any set S whether S is in S. Then if I define a 
> set R s.t. for any set S, S is in R iff S is not in S, I'm defining 
> R only by reference to objective facts: for any set S it should be 
> a fact either that S is in R or that it isn't.
> So, on what not ad hoc, not a posteriori, grounds could I reject R? 
> How could it be ill-defined? How could it not exist?

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