[FOM] Motivating the concept of a generic filter

Timothy Y. Chow tchow at alum.mit.edu
Wed Oct 10 12:29:26 EDT 2007

Jay Sulzberger wrote:
> Ah, because in algebraic geometry, and in the theory of equational and 
> Horn classes of models, and, even, as Boole points out, in the case of 
> finite probability algebras, there is always a generic 
> point/model/probability algebra.  And perhaps, seemingly, naively, there 
> are other examples.  In all these cases (the Boole case is not well 
> known) the generic thing may be described as the model which satisfies 
> all "sentences" of a "theory" and no more sentences.  One does have to 
> be careful in the Boole case.  Thus in these examples, the concept of 
> generic point, most free algebra, Boole's probability algebra given by 
> generators and relations, is seen to be, each, a special case of the 
> general concept "the generic thing".  So one might conjecture, the 
> concept of a "generic set" might also be well defined, in a way very 
> like the earlier definitions.

Can you be more explicit?  For example, can you write down a detailed 
dictionary that gives exact parallels between a generic (ultra)filter and 
these other "generic" objects?  And on the basis of that analogy, what 
would you predict would be the result of imposing the "generic" condition?  
Remember, modding out by *any* ultrafilter yields a model of ZFC.  Why 
would you then be led to consider generic filters in your quest to prove, 
say, the consistency of ~CH?

Tim Chow

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