Talk:Forcing (mathematics)

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I'm a bit new to the guidelines for formatting math in WP, so forgive me if this isn't in line with the policies. I don't have the symbol &#x22a9. Seems like here would be a good place to use <math>, especially since this is a critical symbol and I can't tell what it is.--Luke Gustafson 08:08, 21 December 2005 (UTC)

Which browser are you using? I have edited the article to use the LaTeX symbol, but generally this is not preferred for inline symbols. Have you seen the same problem on other browsers and computers? - Gauge 23:54, 7 January 2006 (UTC)

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Since this is a pretty bulky wiki, I propose to delete the stuff on finding a generic filter that's already in the Rasiowa-Sikorski lemma, namely:

The existence of such a G is follows from the Rasiowa-Sikorski lemma. In short, because M is really countable, one can enumerate the dense subsets of P in M as D0, D1, …. (In general, M thinks the number of such dense subsets is uncountable, and hence this enumeration is not in M. This is a version of the Skolem paradox.) By assumption, there exists p0∈D0. Then by density, there exists p1≤p0 with p1∈D1. Repeating, one gets …≤p2≤p1≤p0 with pi∈Di. Then G={ q∈P : ∃i, q ≥ pi } is a generic filter. In fact, slightly more is true: given a condition p ∈ P, one can find a generic filter G such that p ∈ G. This holds because one can start with p0 ≤ p.

Any objections? Baarslag 22:12, 17 July 2005 (UTC)

I agree, it might as well go, it's redundant with Rasiowa-Sikorski lemma and sort of interrupts the flow of the article.--Luke Gustafson 08:08, 21 December 2005 (UTC)

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I am interested in adding a lot of basic graduate level set theory to WP. The existing articles on forcing, constructibility, large cardinals and the like strike me as terribly lame, mere conceptual stubs. On the other hand, this stuff is not trivial, and perhaps people don't want it on WP. What I've written is a first draft, perhaps too hard, perhaps too much notation, in other words, perhaps not worth it. In particular, at the moment the article is in an internally inconsistent state. It's unclear to me whether I should go forward or not.

So I'll wait for comments.

If the consensus is it's too much, I'll accept that and even do the revert myself.

But I think I can provide a good outline here, and even summarize other various famous results with decent outlines of the proofs, from Martin's axiom to the Solovay model and more--192.35.35.36 19:49, 1 Mar 2005 (UTC).

There is no reason why such material should not be here. I've written some stubby things on set theory, and they're stubby because my energies have been devoted to other subjects; I'd have to re-learn the subject. (In very elementary "naive" set theory, I've contributed to Cantor's theorem, Cantor–Bernstein–Schroeder theorem, and others.) Michael Hardy 22:23, 1 Mar 2005 (UTC)

What I'm most uncertain about is if I'm walking over a general outline that serves a useful purpose that I'm incapable of seeing. My reaction is "no way", "lame", etc., here, and in related articles, and I have been noticing a proliferation of very high level stubs and red links for topics which really can't have a low level description.

Anyway, I'll wait a day or two and see if there is more feedback.--192.35.35.34 22:48, 1 Mar 2005 (UTC)

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I've expanded things, and I hope it is clearer. More to come in a few days.

In the long run, I expect it best to break this up into smaller articles. Presumably c.c.c. and the delta-system lemma (implicit in my write up) deserve their own pages, along with cardinal collapsing. In fact, every forcing poset should have its own page in the end. But for now, I think it best to keep everything on one page. (That is why I've decided to keep Boolean-valued models here for now.)--192.35.35.36 17:24, 3 Mar 2005 (UTC)

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Aleph4: the most common convention is p ≤ q, regardless of whether you are forcing up or down. So, yes, Shelah typically has later letters stronger, but I'm following the down convention.--192.35.35.35 20:23, 7 Mar 2005 (UTC)

First: I am not questioning the "down" convention. Stronger conditions are smaller -- at least if Boolean algebras are involved, anything else would lead to madness.

I didn't think you were trying to change the convention. Perhaps more papers are written with the up convention, just by counting Shelah's output, but everyone else pretty much follows the down convention, for precisely the reason you mentioned. Since at some point the article is going to directly compare forcing with BAs, there's no choice at the WP-level but to follow the down convention.

In fact, Shelah has been known to cheat. In PAIF he was once doing something so complicated, he tells the reader just to take the appropriate "Boolean" combination of conditions, despite that fact that the algebra is upside down.

But in the current article (mainly written by you, it seems; great job!), in most cases the weaker condition is called p, the stronger is q or r. (Look at your definition of "generic filter", for example. This is quite natural, because -- as you undoubtedly know -- most proofs proceed along the lines of

"Let p be an arbitrary condition, then we can find a stronger q deciding bla. ... We can now strengthen q even further to r such that..."

Or, slightly more formally: take any lemma that you prove in a forcing argument, write it in prenex form. If the matrix asserts that p is stronger than q, for two variables p and q, then almost always p will appear after q in the string of quantifiers. Which means that it is more Special:Watchlistnatural (linguistically of course) to choose the names in the opposite order.

For sure! In the middle of a forcing proof, you will typically go to the next letter in the alphabet when you need a stronger condition. But when it comes to first defining the forcing, it is most natural to pick p on the left, and q on the right, with the inequality in-between whichever it happens to be. This is what Jech and Kunen do.

I don't think there will be very many forcing arguments at the level where taking stronger conditions matters. The proofs will all be outlines, right? The generic filter will be looked at, the implications of it existing will be followed, I will try to name names of interesting objects, in short, the issue shouldn't come up.

Anyway, it is not a big deal... -- Aleph4 21:24, 7 Mar 2005 (UTC)

Given that I've got a bunch of expansion lined up, I think convention tweaking should be saved until it's clear what's best for the upcoming article as a whole, when it's time to split it up. For an example of how I'm thinking, when it comes to the Suslin hypothesis, I will kill Suslin lines directly, using their topologies as the posets, with no mention of the trees. My write-up on measure forcing has been influenced, obviously, for the sake of the Solovay model down the road, but also for its similarity to this way of killing Suslin lines. (Please don't tell me its "original research". It's way too trivial to publish separately, and just somehow never got noticed by the textbook writers when running on historical inertia. As far as I'm concerned, it's change-of-notation.)--192.35.35.34 22:57, 7 Mar 2005 (UTC)

Btw, I don't think your definition of the name of the random real is correct. If I would write the text, I would use a name for its Dedekind cut, something like , but apparently you want to really work in 10^ω... -- Aleph4 23:55, 7 Mar 2005 (UTC)

It's effectively (read, intuitively) equivalent to giving the decimal expansion of the random real. It's my experience that the constructive approach is easier to understand (not just me, but most people), and if you can't actually give the constructive approach, then mimicking it goes a long way towards comprehensibility. When it comes to actually doing the mathematics of random reals, the existing approaches in terms of what the generic filter converges to is certainly more efficient.

Also, this approach in terms of intervals will be very similar to the forthcoming Suslin line topology self-destruct argument. Instead of decimal based intervals, there will be maximal antichains of open intervals. The one interval which is in the generic filter gets split by a countable but non-dense subset, giving more intervals. Now iterate ω₁ many times. See the similarity?--192.35.35.36 14:53, 8 Mar 2005 (UTC)

Still not quite right. You mean ((n,k) ̌, [...]), I guess. Assuming that "decimal expansion" is a shorthand for "infinite sequence of decimal digits", and not "nested tower of decimal intervals".

Well, I consider "decimal expansion" shorthand for whatever fits the given situation. That's probably a poor attitude on my part, but it is one of those terms we all learn way before we are ready for formal definitions. As it is, I want to keep it in nested interval form, since it's the simplest. The generic filter literally contains the "decimal expansion", so to speak.

If you identify reals with sets (not sequences) of nested decimal intervals, the name should be { (Eˇ, E): E is of the form [k.10^-n, (k+1).10^-n]} -- Aleph4 10:22, 10 Mar 2005 (UTC)

Well, duh, me. Of course. Actually, that's much better, since it looks almost identical to the canonical name G for G.--192.35.35.36 13:50, 10 Mar 2005 (UTC)

Btw, my default font does not show the spacing character (u+2005) correctly, and in fact it obscures the "check" (u+030c) following it; this is much worse than having the check a little bit off-center. (I assume that if my browser displays it incorrectly, then there is a positive measure set of other users with the same problem.) -- Aleph4 18:19, 8 Mar 2005 (UTC)

How does xˇ look to you? That's x followed by a non-combining hacek. I'll change them, if this is satisfactory.

Looks fine to me -- Aleph4 10:22, 10 Mar 2005 (UTC)

Actually, when I use IE 6.0, much of the Unicode is messed up. Switching the encoding doesn't seem to make a difference.--192.35.35.35 20:04, 8 Mar 2005 (UTC)

Some things need adding:

Some points (which are maybe obvious to everyone else, but confused the hell out of me when I studied this material):

• It needs to be clear that (mostly for useful applications) G \not \in M
• Also M \neq M[G]
• the notion of absoluteness needs to be metioned a bit more to give the flavor
• be clear that c.c.c. is not absolute!

Another thing that I found very useful was seeing forcing conditons as being analogous to Cauchy sequences forcing real numbers into existence from the rationals, using e.g. the surd symbols as things to compose names with. I don't know how rigourous this analogy can be made, (and thus how appropriate it is) but it certainly helped me.