[FOM] CH and forcing

Aatu Koskensilta Aatu.Koskensilta at uta.fi
Mon Jul 4 14:57:06 EDT 2011

Quoting Andreas Blass <ablass at umich.edu>:

> If one insists on two-valued models, the assertion is still correct,  
> provided one does not require models to be well-founded.

   It's standard in expositions of forcing to restrict one's attention  
to well-founded models. Since all the necessary inductions and  
recursions involve only definable predicates this is clearly an  
overkill; all we need is that the ground model satisfied these and  
those induction and recursion principles (e.g. regularity given  
sufficient amount of replacement or whatnot). And, as those in the  
know very well know, we can force over e.g. uncountable models and  
obtain (sometimes necessarily) ill-founded models, just by  
boneheadedly going through the formal motions.

   Is there in the literature any systematic, clear, motivated,  
account of forcing covering all this? My personal experience is that  
everyone knows these things, but I'd be hard pressed if someone asked  
for reference.

Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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