Thanks very much!

]]>I am writing a text, but it is not yet ready.

Eventually it will appear on my substack, Infinitely More

Another observation is that you can permit the bounds that you used in quantifier elimination to be variables, which establishes a nice connection with Presburger arithmetic, see, in particular, the beginning of Section 6 here:

Viktor Kuncak, Huu Hai Nguyen, and Martin Rinard. Deciding Boolean Algebra with Presburger Arithmetic. Journal of Automated Reasoning, 36(3), 2006.

http://lara.epfl.ch/~kuncak/papers/KuncakETAL06DecidingBooleanAlgebraPresburgerArithmetic.pdf

Another nice connection of viewing elements as singleton sets is that you can use results on Boolean algebras to show decidability of monadic class of first-order logic, which in turn is related to set constraints, see this paper that also clarifies that the result dates to Loewenheim 1915 and simplified by Ackermann in his book:

L. Bachmair, H. Ganzinger and U. Waldmann, “Set constraints are the monadic class,” Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science, Montreal, QC, Canada, 1993, pp. 75-83, doi: 10.1109/LICS.1993.287598.

]]>For classical logic, this is straightforward, since validity is defined as being true in every 2-valued model. And every Boolean algebra-valued model has 2-valued quotients as I explain in the post, showing that any statement with a nonzero Boolean value is true in some 2-valued model. For the intuitionistic result, I’m sorry, but I refer you to the constructivists.

]]>Any chance you may have a source to recommand? ðŸ™‚

]]>Any chance you might have a source to recommend ?

]]>Yes in both cases.

]]>That’s a very good question. I strongly suspect that it is not equivalent, but I don’t have a countermodel at hand.

]]>Thanks for these comments. Indeed, I had meant that there is no formula df(x) that picks out the definable elements in every model M. In some models, the definable elements do form a definable class, such as any pointwise definable model. And indeed, the argument generalizes to other theories, not just ZFC, provided that there are pointwise definable models. (Not every theory with infinite models has a pointwise definable model.)

]]>Looks like there is a presentation now.

]]>Though, why do you use ZFC there at all since presumably the claim holds for any theory T having infinite models. Right?

]]>The point about nonfunctoriality is, the generic you build depends on your enumeration of the model. So running the process on isomorphic copies of the same model of set theory may not produce the same generic. Or phrased in another way: forcing isn’t determinate in just the model, but it is determinate in the model equipped with an enumeration of its elements.

I’m not sure whether this undercuts your proposed rebuttal to forcing potentialism not being real potentialism. Presumably the forcing (and only forcing) potentialist wouldn’t think their models are countable, so they wouldn’t think it makes sense for them to come along with an omega-enumeration.

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