[FOM] The semantics of set theory

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Sun Oct 6 12:38:58 EDT 2002

From: Ralf Schindler <rds at logic.univie.ac.at>
On Thu, 3 Oct 2002, Kanovei wrote:

> Generally, there is no way to define ZFC-truth other than to 
> extend the language of ZFC. 
> Three typical methods are known. 
> Third, consider a second-order impredicative theory of classes. 

One doesn't need an *im*predicative theory of classes here, one can
do with predicative classes. (A class is predicative iff it can be
defined by a fmla of set theory + parameters for sets.) --Ralf

To define that a set theoretic formula A (with parameters or 
even without parameters) "is true" one has to claim the 
existence of a class satisfying certain known properties 
and containing A. 
Such a class itself cannot be definable, e.g. predicative, 
if we want to treat A as a free variable. 
Therefore, in this case, predicative classes do not suffice. 

If, on the contrary, we are going to consider A(x_1,...,x_n) ^?^[[3~
as a fixed, metamathematically given, formula, then prediva^[   
predicative classes suffice, but the whole problerm results in 
the tautology : "A is true" is replaced by A. 


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