Philosophy of forcing

Question

There are usually two expository styles on forcing: internal (forcing over the universe $V$) and external (forcing over a ctm $M$); I guess whether to use general poset $P$ or boolean completion $B$ is non-essential. In the first approach we get a boolean-valued model, and in the second a larger ctm $N\supseteq M$. Any way, forcing does show the independence of some statements such as CH, but-if I am a platonist who believes there is an "absolute" or "real" universe of sets-what does forcing tell me about the universe?

I am asking this question because many people seem to agree that if something is easy or even possible to force then it should be true in the real universe. For example, in the paper Believing the Axioms I, section 2 the author lists various arguments in favor of/against CH, the last of which says that it is much easier to force CH than not-CH, so it should be true. We also have the popular axiom PFA; I know very little about large cardinals, but it seems the consistency of PFA is usually proved by forcing, and not many large cardinal axioms directly imply PFA; nonetheless PFA is believed to be true by many people. Why does a ctm or a boolean-valued model that satisfies PFA indicate its truth in the real universe?

On the other hand, let's say I am more flexible and not a platonist. When I do forcing, say Cohen, am I playing formal games with ctms/boolean-valued models, or am I really imagining adding a new set to the current universe? If it's the latter, how do I convince myself of the existence of generic filter $G$? Also, since any cardinal can be collapsed by forcing, would I have to admit that all sets are countable and uncountability is only an illusion (Dana Scott seems to suggest this in his foreword to the book Set Theory: Boolean-valued Models and Independence Proofs)?

Some other random questions: I remember reading that there does exist a point of view that we can never reach the whole universe $V$, and any model (not necessarily countable) can be enlarged to one with more reals, but I can't find the reference. I have heard of Hamkins's multiverse point of view, but I am not sure if I am ready to accept that every model is ill-founded...

Answer 1

If you believe that there is a single universe, then forcing tells you absolutely nothing about its truth.

It does tell you something about the limits of what you can prove. If you only want to assume that the universe satisfies $\sf ZFC$, then forcing tells you that you cannot determine if $\sf CH$ is true, or if there are Suslin trees, or if Martin's Axiom is true.

Forcing lets you calibrate your view as to what should be true by virtue of what can be forced, or "how easy it is to force a statement" using some subjective measure of "easy" (arguably it's easier to force $\lnot\sf CH$, since you just need to add Cohen reals with a ccc forcing; and arguably it's easier to force $\sf CH$ since you can always do that without changing the real numbers by adding a bijection between $\omega_1$ and $\Bbb R$. Both arguments are equally good/bad).

If you are not a platonist, then how you view forcing is really up to the way you prefer to think about it. In my first paper I talk about countable models and so on, because I wasn't yet comfortable with the machinery of forcing, but in later papers this is not the case anymore. Nevertheless, some set theorists who are far more experienced than me still talk about countable models, because that is how they view forcing. It is a personal thing, and as long as you understand, at least in theory, why all the approaches are the same, mathematically speaking, you shouldn't have any problems.

Answer 2

A little while ago, I wrote a paper addressing some linked worries to what you have here:

Barton, N. Forcing and the Universe of Sets: Must We Lose Insight?. J Philos Logic 49, 575–612 (2020). https://doi.org/10.1007/s10992-019-09530-y

I'll add a very brief summary of what I argue there:

1. There's pressure to want to provide "nice" interpretations of forcing, since forcing is more than just a tool for proving relative consistency. We can also formulate axioms about uncountable sets using forcing (e.g. remarkable cardinals) and prove theorems about uncountable sets in the universe using forcing (cf. Todorčević and Farah's book, Malliaris and Shelah's result that $\mathfrak{p}=\mathfrak{t}$).

2. You can get "nice" interpretations of forcing within the universe, where the model you use is very "close" to $V$ in certain senses.

Regarding truth and forcing, a great example here is Bagaria who has written several articles on the idea of bounded forcing axioms, which roughly state that if you have a certain kind of set in a forcing extension, then you can find one in the universe.

[Bagaria, 2005] Bagaria, J. (2005). Natural axioms of set theory and the continuum problem. In Proceedings of the 12th International Congress of Logic, Methodology, and Philosophy of Science, pages 43–64. King’s College London Publications.

[Bagaria, 2006] Bagaria, J. (2006). Axioms of generic absoluteness, pages 28–47. Lecture Notes in Logic. Cambridge University Press.

[Bagaria, 2008] Bagaria, J. (2008). Set theory. In The Princeton Companion to Mathematics, pages 302–321. Princeton University Press

Tight controls are needed to get everything consistent with $\sf ZFC$. (Observe that both $\sf CH$ and $\sf \neg CH$ are $\Sigma_2$ and so you'll need some restriction on the formulas allowed. Parameters are also dangerous in the $\sf ZFC$-context, since you can collapse $\omega_1$ or some other uncountable set in an extension.) In the end though, if you want to have a principle that anything forcable is true, you're going to be pushed in the direction of every set being countable. (At the risk of over-plugging my own work, I've written on this too: "Countabilism and Maximality Principles" https://philarchive.org/rec/BARCAM-5.)

Oh and regarding "multiversism" in the presence of well-foundedness see:

Steel, John, 2014, “Gödel’s Program”, in Interpreting Gödel, Kennedy, J. (ed.) Cambridge: Cambridge University Press, 2014.

Meadows, Toby, Naive Infinitism: The Case for an Inconsistency Approach to Infinite Collections.

Scambler, C. Can All Things Be Counted?. J Philos Logic 50, 1079–1106 (2021). https://doi.org/10.1007/s10992-021-09593-w

Arrigoni, T, and Friedman, S.-D. (2013). The Hyperuniverse Program. The Bulletin of Symbolic Logic, 19(1), 77–96. http://www.jstor.org/stable/41825307