Is forcing force on us

[Giorgio Venturi]

Dear Tim,

thanks for the interesting question on the universality of forcing. I have
a couple of comments on extensions and restrictions of ZF, that might help
the discussion.

The first one is connected to a recent application of Robinson model
theoretic forcing to the models of ZF, that have shown interesting
connections with generic absoluteness (
https://link.springer.com/article/10.1007/s11225-019-09851-8). It has been
recently proved (https://arxiv.org/pdf/2003.07114.pdf) that, modulo a
suitable extension of the language of set theory, the \Pi_2 theory of
H_\aleph_1 is the model companion of ZFC + large cardinals. This result is
relevant for this discussion because it shows that for a class of sentences
(\Pi_2 sentences with parameters in H_\aleph_1) forcability and consistency
are equivalent, in the presence of large cardinals (it can also be proved
that Forcing Axioms have an analogous role with respect to H_\aleph_2
https://arxiv.org/abs/2003.07120). Therefore, for some extension of ZFC and
with respect to specific classes of sentence, forcing is really the "only"
method available.

The second comment concerns a clarification of what we might mean by
weakenings of ZF. Indeed, we might try to weaken the non-logical part of
the theory, as it was probably meant in Tim's clarification of his
question, or we might weaken the logical part. The difficulty with this
second choice consists in showing, first, that ZF is compatible with weaker
logics and, then, that these non-classical versions of ZF can give new
consistency proofs. In this case I have a positive answer: there are
non-classical versions of ZF, based on logics weaker then classical or
intuitionistic logics (see
https://link.springer.com/article/10.1007/s11225-020-09915-0 for an example
of such), for which we can construct algebra-valued models (generalizations
of Boolean-valued models, where the algebra used to evaluate the sentences
is not necessarily Boolean) that validate sentences that are just false in
classical ZF (the relevant paper is currently under review). I am not
claiming that this shows that (classical) forcing is not the only method
available. Indeed, there is a lot to accept in going non classical.
However, this shows that depending on how liberal we are with respect to
semantics, we might get interesting and possibly incompatible answers to
Tim's question; even keeping with the (non-logical) axioms of ZF.

Best,
Giorgio

Em sáb., 22 de ago. de 2020 às 01:44, Timothy Y. Chow <
tchow at math.princeton.edu> escreveu:

> On Fri, 21 Aug 2020, John Baldwin wrote:
> > What is the distinction between arithmetic and set theory that makes
> > forcing possible?
>
> I should clarify that I had in mind potentially "exotic" weakenings of ZF
> that may not have been studied much.  In other words, I'm imagining
> tinkering with the ZF axioms specifically to see what is forcing us to use
> forcing (as opposed to tinkering with the axioms to create systems of
> intrinsic interest).
>
> But as for your question here, I tried asking something similar on
> MathOverflow a while ago.
>
> https://mathoverflow.net/q/100792
>
> Tim
>
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