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December 16

Forcing -vs- Boolean Valued Models

I have books on both - and my understanding is they are equivalent - but I wanted to ask which is better to study, in detail? I have a basic understanding of both, and intend to go into both in detail, but wanted a better idea of which is "more useful" in terms of seeing connections and understanding set theoretic things - as opposed to, say, actually proving some specific result. Our article on forcing indicates BV models, but I'd be curious to hear actual input to get a clearer idea of things. (Thank you for any answers, or help, the further I go, the harder it is to keep a clear sense of direction, all of you are immensely helpful and I am truly appreciative for all the help I've received in the past).Phoenixia1177 (talk) 15:07, 16 December 2014 (UTC)[reply]

I can't help but I'll point out that this is not about forcing in dynamical systems (e.g. [1]), but rather Forcing_(set theory). One might suspect that there are some analogies between the concepts based on the name but even that is unclear to me. SemanticMantis (talk) 19:17, 16 December 2014 (UTC)[reply]

This is a bit of a matter of opinion, but here's my take on it: Forcing is usually a more efficient way to actually make progress than Boolean-valued models are. A forcing partial order can be described in such a way that you can see "what it's trying to get at", what sort of generic object it's trying to add to the universe. The corresponding Boolean algebra is usually a lot less perspicacious. If you have to back off all the way to the general case (the algebra of regular open sets in the topology generated by cones from the partial order) then you've basically lost all useful intuition about what the elements of the Boolean algebra are.

However, it's very nice to know the Boolean-valued model approach for other reasons, to be able to have a more philosophically satisfying interpretation of certain results. For example, take the the result about making

2^{\aleph _{0}}

equal to κ, for a given uncountable regular cardinal κ. What does that mean?

You can't capture it completely in proof-theoretic terms ("adding the assumption

2^{\aleph _{0}}=\kappa

to ZFC cannot produce an inconsistency in ZFC unless there was one already") because κ is just some random cardinal, one that may or may not even be definable in the language of set theory, so there is no way to add

2^{\aleph _{0}}=\kappa

as an axiom. It's a category error even to try.

Similarly, you can't get at it in any obvious way via countable transitive models (the usual fallback for talking about forcing in terms of concrete objects) because κ is not an element of any countable transitive model. There may be some way to make sense of it using countable transitive models that are Mostowski collapses of models containing κ; not sure on this point.

But with Boolean-valued models, there's a very clear interpretation. There's a Boolean algebra B such that the statement

2^{\aleph _{0}}=\kappa

has probability 1 in

V^{B}

. --Trovatore (talk) 19:44, 16 December 2014 (UTC)[reply]

Thank you very much for your insight, as always - I think I am going with BVM's first, then:-)Phoenixia1177 (talk) 11:12, 19 December 2014 (UTC)[reply]

Hmm, maybe I buried the lede on that a little bit. My more important point was that if you want to understand a given forcing notion (Cohen forcing, random-real forcing, Prikry forcing, what have you), then it's much more clear to think about the partial order than the Boolean algebra. What BVM's give you is more of a nicer story to tell yourself about what the final results mean. --Trovatore (talk) 14:44, 19 December 2014 (UTC)[reply]

That's how I took it - but, oddly, having the ability to tell nicer stories about results, first, will make forcing easier to comprehend for me. My struggle has never been with understanding the formal/mathematical aspects of things, but putting them in a coherent context and having an intuitive sense of meaning. So if BVMs make that easier, then having that context will make the more useful notion (forcing) simpler to learn and use. I end up reading a lot of things "out of order", so to speak, for this reason largely. (As an aside: your example is fascinating and reminds me of a discussion on the talk page for the Continuum Hypothesis involving Easton's Theorem and what 2^c "can be".) --unless you mean that learning BVMs requires already knowing forcing - the books I'm using don't seem to indicate that, but I've gotten a few chapters out of the way, so I may not have hit that hurdle.Phoenixia1177 (talk) 17:13, 19 December 2014 (UTC)[reply]

Well, sure, except I think you're going to hit a dead end pretty quickly trying to understand forcing/BVMs in the abstract, without examples. And as soon as you're talking about particular forcing notions, it's a lot easier to gain intuition if you don't have to deal with the overhead of the stuff you have to add to the partial order to make it a Boolean algebra. --Trovatore (talk) 20:35, 19 December 2014 (UTC)[reply]

That makes a lot of sense - I've had that happen with jumping around before and ended up reading side by side, which was a pain, but doable (other subjects). Do you have any good readable books you might recommend to get started with Forcing? I've found a few on BVMs that seem reasonably readable, which was part of the motivation to go that way first - I'm less sure where to get started with Forcing.Phoenixia1177 (talk) 05:00, 20 December 2014 (UTC)[reply]