[FOM] CH and forcing
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
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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