Timeline for A better way to explain forcing?
Current License: CC BY-SA 4.0
25 events
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| Aug 24, 2020 at 0:46 | comment | added | Asaf Karagila♦ | @Timothy: Easy, then, use random reals. | |
| Aug 24, 2020 at 0:35 | comment | added | Timothy Chow | @AsafKaragila : I'm still thinking it might be a useful crutch for the beginner. For $\neg$CH, it's easy to motivate adding a function from $\aleph_2^M \times \aleph_0$ to $\{0,1\}$ and it is easy to say what it means to add a random function. No need to define "filter" or "dense" or "generic". Forcing then means "almost surely implies." So this allows one to skip some annoying definitions and get to the main point more quickly. The downside is then you may have to "unlearn" the randomness terminology later. | |
| Aug 23, 2020 at 10:34 | comment | added | Asaf Karagila♦ | @Miha: The terminology of "generic" is a bit of an odd duck. Cohen did use it in the sense of "typical" or "random", but once the topological and Boolean-valued approaches were starting to clarify, the term "generic" was taken from the more standard sense of "meeting dense sets". And of course, that fits Cohen's use, but only because the term "generic" in topology/algebraic geometry came from that same sense of the word. I agree that changing "generic" to "random" seems like a confusing reason why forcing would suddenly make sense. | |
| Aug 22, 2020 at 16:51 | comment | added | Mirco A. Mannucci | Correction: replace "function" with "set" in " if I choose a function outside M which does the job" . | |
| Aug 22, 2020 at 16:21 | comment | added | Mirco A. Mannucci | I have just eaten my italian risotto, so I am not especially lucid right now, but my sense is that M{G1] and M[G2] are different models, and yet they agree as far as the job they need to accomplish. Perhaps this is a key: G would then be generic in the sense that it does not matter at all which one I choose... | |
| Aug 22, 2020 at 16:18 | comment | added | Mirco A. Mannucci | on the other hand, let us say you have a P inside M, or equivalently a boolean algebra B. ANY ultrafilter will squeeze the boolean model to a "real" model, so in this respect it looks as if there are many possibilities. It would be interesting to ask oneself: suppose I choose TWO ultrafilters G1 and G2, what is the relation between M{G1] and M{G2}? | |
| Aug 22, 2020 at 16:15 | comment | added | Mirco A. Mannucci | @TimothyChow the last two points you touched upon are, in my modest opinion, both great, but will require some serious thinking to be put to use. Let us start from the first one, namely "generic" as " likely to be a non-virus" ie something that can be safely added to M without causing problems. I may be wrong, but to me it feels as if the situation is just the opposite: if I choose a function outside M which does the job, chances are it WILL mess things up. So, Cohen's methods seems to be: I wanna be as conservative as possible in choosing G, so that the chances of creating trouble are zero. | |
| Aug 22, 2020 at 16:13 | comment | added | Asvin | Yes but I only learnt about graphons very recently. Your point about category vs measure is very relevant and the Rado graph is "easier" because it's a 0-1 thing. For graphons, they are indeed very nice but I would be to think a little more about them before I can say more! | |
| Aug 22, 2020 at 16:08 | comment | added | Timothy Chow | @Asvin : Do you know about the Rado graph and about graphons? As an aside, in analysis there is a distinction between measure and category ("category" here means Baire category and not morphisms/functors). Generic filters are closely related to category whereas probability theory is related to measure. So one has to be a bit careful about being too literal in one's identification between "almost all" and "generic." | |
| Aug 22, 2020 at 7:44 | comment | added | Asvin | This is really an aside but I have been thinking lately that maybe we can have better foundations for random variables inspired partly by this forcing stuff (and algebraic geometry). In both places, we want to use generic as saying "almost all" and that is also what the phrase "with probability one" means. However, in AG/Forcing, it seems that having a generic variable is stronger than just thinking of "with probability one" - you can do operations on the generic object directly. Can something similar work in, for instance random graph theory where a random graph would be an actual graph!? | |
| Aug 22, 2020 at 7:17 | history | edited | Greg Martin | CC BY-SA 4.0 |
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| Aug 21, 2020 at 16:19 | comment | added | Timothy Chow | @MihaHabič : The difference may be mainly psychological, but for example, standardly, only G gets the adjective "generic." Stuff that depends on G doesn't. But if X is a random variable, variables that depend on X are also random variables. They inherit the randomness. But I don't want to get into a semantic debate. It's just an idea that switching terminology might introduce a new perspective. E.g., is avoiding contradiction an excruciatingly delicate process, or is it easy (and it's just the consistency proof that's hard)? | |
| Aug 21, 2020 at 16:06 | comment | added | Miha Habič | @TimothyChow To the point of switching from "generic" to "random", don't they mean essentially the same thing? Cohen started with Cohen forcing over a countable model, where the generic filter corresponds to a real in a particular comeager set. In other words, any real from that comeager set would have worked for his argument, so a "generic" (aka "typical") real works. It was always my impression that this is where the terminology comes from. Random reals work the same way but with measure one sets. | |
| Aug 21, 2020 at 12:54 | comment | added | Timothy Chow | Of course I'm still piggy-backing on all the usual machinery to prove that a random function works, but still, it seems suggestive. The mentality has switched from, "We have to be really really careful to avoid a contradiction" to "The burden of proof is on the contradiction to manifest itself." Is this mentality accurate? If so, can it be pushed further? | |
| Aug 21, 2020 at 12:51 | comment | added | Timothy Chow | Cool! By the way, let me mention an idea/question of Scott Aaronson's. Instead of the word "generic" let's try using the word "random." We know that some G might create contradictions. But it seems that producing a contradiction is actually a delicate process. It won't happen at random. So just pick a random G, and with probability 1 (or least with some positive probability) everything will be fine. I think this idea works for $\neg$CH. If I pick a random function $F:\aleph_2^M\times \aleph_0\to\lbrace0,1\rbrace$ then the associated filter will be generic. | |
| Aug 21, 2020 at 12:36 | comment | added | Mirco A. Mannucci | @TimothyChow just added an addendum on that one | |
| Aug 21, 2020 at 12:35 | history | edited | Mirco A. Mannucci | CC BY-SA 4.0 |
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| Aug 21, 2020 at 12:08 | comment | added | Timothy Chow | I am reminded that in Shelah's "Logical Dreams," one of his dreams is to "show that forcing is the unique method in some non-trivial sense." | |
| Aug 21, 2020 at 9:52 | comment | added | Mirco A. Mannucci | forcing is in a sense to be made precise inescapable here. Anyway: how about launching a plan to investigate the issue of "virus information"? For instance, suppose I have a weaker set theory, how does it make less likely that an external G would mess things up when added? I suspect that without full blown replacement and power set there would be less trouble to think about... | |
| Aug 21, 2020 at 9:47 | comment | added | Mirco A. Mannucci | I hear you man. The key issue is of course that whatever G you add to M, it can carry some "hidden information" which is an obstruction toM[G] being a model. Now things become slippery: what kind of information? Unfortunately ZF is very complicated, and it is not trivial to determine a priori which kind of information G can bring in. For instance, you have replacement, so perhaps G alone can seem pretty harmless, but once it is inside it can be used to define a new well ordering. My lingering feeling is this: unless there is a totally NEW way of building models, | |
| Aug 21, 2020 at 0:59 | comment | added | Timothy Chow | This is a nice account of why forcing works, but I guess I'm trying to ask why something more simple-minded doesn't work. Certainly, pre-Cohen folks can't be faulted for failing to find G, describe it, and add it. That was a hard problem. But Cohen found G and described it. With the benefit of hindsight, why can't we simplify the argument? What is it about the ZFC axioms that seemingly compels us to use such elaborate machinery? | |
| Aug 20, 2020 at 23:17 | comment | added | Mirco A. Mannucci | @PaceNielsen, thanks for the appreciation! Actually, I have just learned something from you: indeed forcing belongs to the same order of ideas you mention. It is not by chance that Fitting, formalizing its logic, found out that it is "intuitionistic". Writing this answer helped me to clarify to myself a lot of things: for instance, following your lead, how do you complete a category? Assume the job has already being done, now look back to you old structure. The new guy leaves "traces" in it. The idea is that by patching these finite traces you will assemble what you need... | |
| Aug 20, 2020 at 23:04 | comment | added | Pace Nielsen | I've found that that model of construction (having little parts, and gluing them together into a whole, according to some instructions) to be a very useful paradigm for solving problems. The additional idea that some infinite processes/concepts keep some of the properties inherent in finite processes/concepts, is also quite useful. (This happens, for instance, in products and coproducts in category theory, all the time. The compactness theorem is another instance of this principle in action.) | |
| Aug 20, 2020 at 21:50 | history | edited | Mirco A. Mannucci | CC BY-SA 4.0 |
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| Aug 20, 2020 at 21:44 | history | answered | Mirco A. Mannucci | CC BY-SA 4.0 |