Forcing: cardinals below c · deprecated: mathematics

I'm trying to understand forcing, and this post by Aaron Scott got me started once again:

Ever since then I've been wondering if we can describe or interpret one of the cardinals between

\aleph_0

and

\beth_1

Forcing is inherently nonconstructive, so it's not at all obvious what the answer should be. I think that these are relative to the model of ZFC we're looking at.

I've been wrestling with Cohen's forcing again. And I have questions. Here's what got me restarted into this topic, and much of the following is inspired by it: Scott Aaronson: The Complete Idiot’s Guide to the [...] Continuum Hypothesis: Part 1 https://www.scottaaronson.com/blog/?p=4974

- Refurio Anachro (@RAnachro)

Basically, I claimed that there are smaller topoi, like the effective topos, where everything is computable, and that that example limits the number of reals to

\omega_{CK}

. But I grew suspicion that I'm doing it wrong! I just don't know enough about these things to be sure! So I came here for help.

By the way, in that thread I said I didn't quite understand König's theoren, but that has changed. Here's a discussion about it:

@silvascientist @CujoUniversity @mathemensch @AndresECaicedo1 @littmath @SC_Griffith I see. Yes, that's nice and crisp way to put it! Looking at König's paper again it now seems crystal clear. He has a rather explanative style, and he surely would have drawn a similar picture had he known Gödel and Cohen's results, but he wrote it in 1904.

- Refurio Anachro (@RAnachro)

John Baez (Dec 27 2020 at 01:39):

Hi! Long time no see. You mean Aaron Scottson, the famous expert on quantum computing. :upside_down: He has promised to explain forcing sometime on his blog, so I'm waiting for that.

Matteo Capucci (he/him) (Dec 27 2020 at 08:51):

My spidey sense are tickling. Is this true? I think the Boolean models/topos theoretic avenues are quite constructive in practice.

Matteo Capucci (he/him) (Dec 27 2020 at 08:52):

Refurio Anachro (Dec 27 2020 at 12:41):

Nice to meet you, @John Baez , it's been too long. Yes, of course, it's Scott Aaronson XD

I linked part one of his 'Complete Idiot's Guide to the Independence of the Continuum Hypothesis', which seems to be just that. He might well cover my questions whenever he gets there, but I just can't sit still for an indefinite amount of time...

Not quite sure why I didn't escalate this to the comment section below his post right away, and I think I should try that later today. Maybe, because there should be some experts on topos theory here, who might be better equipped to spot how silly my gueswork really is. Toposes, i wanted to learn more about toposes. Sigh.

Refurio Anachro (Dec 27 2020 at 12:43):

Thanks, @Matteo Capucci! I might be reading too much into it, but your spidey sense seems to give me some hope that I might not be entirely on the wrong track! You don't happen to have some search terms or even references about this boolean approach to forcing, do you?

Anyways, thanks for the correction of Scott's name, it was clearly too late for me!

Matteo Capucci (he/him) (Dec 27 2020 at 13:04):

:joy: It was really meant to be a correction, more of a funny remark about the double mistake

Matteo Capucci (he/him) (Dec 27 2020 at 13:05):

BTW I've learnt about topos-theoretic forcing from MacLane and Moerdijk book Sheaves in geometry and logic. There's a wonderful section walking through the construction in Chapter IX, iirc. It's the chapter on logic.

Matteo Capucci (he/him) (Dec 27 2020 at 13:06):

Oh also Dana Scott's article on boolean valued models for forcing is very easy to read an very insightful

Matteo Capucci (he/him) (Dec 27 2020 at 13:06):

Refurio Anachro (Dec 27 2020 at 14:14):

Usually, for me, the joy of learning exceeds feelings of frustration I might have when getting corrected. But then again, I don't do papers, my job doesn't depend on it, and it's mostly me having fun on social media. So it may be easier for me to stomach any or rather the eventual embarassment.

Todd Trimble (Dec 27 2020 at 15:24):

I am not in any respect an expert on topos-theoretic forcing. But my memory is that category theorists explain it in three steps: (1) construct a suitable category (often a poset) or site of "forcing conditions", and then take sheaves on that site, (2) pass from there to a Boolean topos by taking double negation sheaves (I'm pretty sure this step is constructive). So far I think this is very similar to consideration of Boolean-valued models. Then (3) collapse to a two-valued model by taking a "filterquotient" along an ultrafilter. Generally, the existence of such a suitable ultrafilter will involve a nonconstructive aspect, unless I'm confused of course. I hope someone more knowledgeable passes by to confirm or refute.

Matteo Capucci (he/him) (Dec 27 2020 at 15:46):

That's exactly right Todd. It is notable though that the last step (which, as you note, is the only non constructive one) isn't strictly necessary. The topos of

\neg\neg

-sheaves on the forcing conditions is already a model of the theory, although a somewhat nonstandard one.

Matteo Capucci (he/him) (Dec 27 2020 at 15:48):

Boolean valued models is what you get when you do the same construction and remove the topos-theoretic framing, only to leave the logical content

Matteo Capucci (he/him) (Dec 27 2020 at 15:49):

In other words, Boolean valued logic can be used to produce a tripos whose associated topos is equivalent to the sheaf-theoretic one

John Baez (Dec 27 2020 at 16:06):

Todd Trimble (Dec 27 2020 at 16:11):

Yes, I've often heard that said, and I myself accept it. But wonder if a traditional set theorist would ever accept that as not cheating or skimping.

Matteo Capucci (he/him) (Dec 27 2020 at 17:02):

Oh yeah, they wouldn't :) in the end, it's a trade-off between being constructive and being two-valued. You choose what you need!

John Baez (Dec 27 2020 at 17:41):

One thing I found a bit frustrating about the treatment of forcing in Mac Lane and Moerdijk's book is that they don't carry it through all the way to proving that there are (traditional) models of ZF where AC or CH fails. Or at least sketching the proof. I guess they didn't want to get tangled up in that stuff, but an application of topos theory to traditional logic would help sell topos theory and maybe also help some of us understand forcing. I haven't read this section in a long time, but at the time I felt the didn't solve the "last mile" problem - you know, like running fiber optic cable to your town but not to your house.

Fawzi Hreiki (Dec 27 2020 at 17:48):

John Baez (Dec 27 2020 at 17:53):

Yes, models of all the ZF axioms, that are models in the sense of traditional set theory - set or class-valued models.

John Baez (Dec 27 2020 at 17:54):

I bet Mac Lane and Moerdijk could have sketched this stuff without sinking into a morass of detail.

John Baez (Dec 27 2020 at 17:54):

But I imagine they thought it was too distracting. Part of the point of topos theory is to avoid a bunch of this messy traditional stuff.

Dan Doel (Dec 27 2020 at 18:02):

Was it even known how to interpret some of the ZF axioms in a topos at the time? The stuff in that book can probably only handle bounded versions of some axioms in an obvious way, no?

Fawzi Hreiki (Dec 27 2020 at 18:09):

The difficulty there though is that you have to re-extract the cumulative hierarchy from the topos of sets which in itself is a whole task

John Baez (Dec 27 2020 at 18:10):

Okay, so maybe it's really hard, or at least tiring. But then they should have said a bit more about why...

Dan Doel (Dec 27 2020 at 18:15):

I guess there were known solutions, of which that is one. But I was thinking of 'stack semantics,' which wasn't.

Dan Doel (Dec 27 2020 at 18:18):

The stack semantics paper also says that the cumulative hierarchy approach requires completeness, which isn't necessarily preserved by filterquotients, so that could be an extra problem. :smile:

Matteo Capucci (he/him) (Dec 27 2020 at 18:29):

:thinking: I'm not understanding your complaint, John. Shall they have talked about the non-topos theoretic way to do forcing? It seems beyond the point!
Also there's a section about algebraic set theory and models of bZF in topoi.

Fawzi Hreiki (Dec 27 2020 at 18:34):

You can deal with ZF models of set theory while still using topos theoretic forcing, but it means you have to also deal with the second aspect too.

John Baez (Dec 27 2020 at 18:48):

No, I was saying I wanted them to say more about how topos-theoretic methods can be used to prove results about traditional logic.

But maybe they said enough and I just wasn't satisfied on my first reading. Let me look. On page 279 they say they'll prove

Theorem 1. There exists a Boolean topos satisfying the axiom of choice, in which the continuum hypothesis fails.

I was wanting them to go a bit further and talk about how to use this to get a traditional model of ZF in which the continuum hypothesis fails.

John Baez (Dec 27 2020 at 18:50):

My point is that to sell topos theory to logicians, it's useful to assure them that topos theory can prove results they're already interested in.

John Baez (Dec 27 2020 at 18:51):

If the problem of translating the topos-theoretic results into ZF is complicated and/or boring, it's fine to offload it to a reference.

John Baez (Dec 27 2020 at 18:53):

(Btw, I had dinner at Charles Fourman's house at CT2019 - he lives in Edinburgh, and he's a well-established computer scientist now. I wish I'd asked him about this stuff - mainly, what it felt like to work on that project.)

Fawzi Hreiki (Dec 27 2020 at 19:01):

Surely though if someone is interested in proving complex independence results about ZFC in particular, they’ll just work directly with the cumulative hierarchy.

After all, there must be a reason why topos theoretic approaches to forcing didn’t take off in the the main set theory circles.

Fawzi Hreiki (Dec 27 2020 at 19:04):

Despite classes not really existing in ZFC, it is in the end really a class theory rather than a set theory. In contrast, the notion of class in ETCS doesn’t make much sense.

Maybe something like algebraic set theory is closer then to how working ZF style set theorists think of the universe of sets (rather than just the topos of sets)

Todd Trimble (Dec 27 2020 at 19:30):

Actually, I think there's probably a good number of set theorists/logicians of the non-categorical sort who are (or say they are) open to learning more about the topos-theoretic approach, but what some of them wonder about is: how is this really different from the Boolean-valued model approach of Scott & Solovay (who are part of the in-club)?

Todd Trimble (Dec 27 2020 at 19:31):

Actually, though, what might be good in the long run is just accept the fact that most set theorists are set in the ways of generic filters and such, and really address other people out there, the young, who might be open to the more 'algebraic' approach afforded by the topos theory methods. In effect, feel free to ignore the traditionalists, until they start showing interest.

John Baez (Dec 27 2020 at 19:55):

One reason is that many traditional set theorists don't understand topos theory. Another reason is that some set theorists consider topos theory a competitor and prefer to "battle" it in places like the Foundations of Mathematics mailing list rather than to learn about it.

But I think in the end set theorists and topos theorists have a lot to learn from each other... and eventually the "battle" will fade away.

John Baez (Dec 27 2020 at 20:05):

That's really the only thing that works. The nLab has a great potential for luring young mathematicians into thinking about logic in more flexible, modern ways.

Fawzi Hreiki (Dec 27 2020 at 22:29):

It's fun sometimes to go into the FOM list archives to see Colin McLarty vs. everyone else but in the end I doubt there were many converts. You're probably right that the soft diplomacy approach via the nLab and the blogs (particularly yours) and this Zulip are more effective.

Refurio Anachro (Dec 27 2020 at 22:43):

Im happy to be ignorant of either side, I just want to know about a model where CH does not hold, and we know what those pesky cardinals in between are and what they mean! Welp.

John Baez (Dec 27 2020 at 22:44):

I know that there are models of ZF where

c

, the cardinality of the continuum, is as big as you like in terms of alephs. Like, it can be

\aleph_{379}

. But I don't understand these models.

John Baez (Dec 27 2020 at 22:45):

I "know" it just in the sense that I've been assured it's true by people I trust.

John Baez (Dec 27 2020 at 22:46):

But anyway, in my example the "pesky cardinals in between" are

\aleph_1, \aleph_2, \dots, \aleph_{378}

Refurio Anachro (Dec 27 2020 at 23:12):

Yes, I know that we can we can make the cardinality of c any cardinal that cannot be written as the supremum of countably many cardinals. That includes

\aleph_1, \aleph_{42}

, but

\aleph_\omega

isn't allowed, due to the theorem of König I mentioned in the OP. Then

\aleph_{\omega+1}

is okay again, and so on. I think the first allowed limit cardinal is

\aleph_{\omega_1}

the next such being

\aleph_{\omega_1•2}

, and so on.

What really interests me is a model that isn't nebulous in a nonconstructive way, such that its

c

is an interesting or otherwise meaningful cardinal with respect to the model. I guessed earlier that maybe the effective topos and

\aleph{\omega_{CK}}

make sense superficially, but I feel that that isn't even compatible with König's theorem, and it's really just a wild guess! I have no way to check it, or even a good reason to actually believe it's true.

John Baez (Dec 27 2020 at 23:15):

I don't know if any of these other models are "constructive" in some sense of that term. That's a fun question that someone should know something about!

On a vaguely related note: no nonstandard models of arithmetic have the operations + and

\times

being computable. So you have to learn to live with that, if you're studying nonstandard models of arithmetic!

Refurio Anachro (Dec 27 2020 at 23:30):

Sure, I'd be happy to jump through all sorts of hoops, just to arrive at a setting where we can look at a model A's c from the outside (from a model B) and make actual sense of it being countable (as seen from B).

You see, people (past me included) misunderstand forcing in the sense that there might be cardinals between

\aleph_0

and

c=2^{\aleph_0}

, but that's not how it works! In fact, forcing extends a small model, where CH holds, by adding new reals, making the earlier c look countable in the new model. At least, that's what Scott seems to say.

There must be an interesting pair of models A, B, where

c_A

is an interesting ordinal in B. (I hope that's not too nonsensical)

John Baez (Dec 27 2020 at 23:33):

Well, in the models where CH fails there are cardinals between

\aleph_0

and

c = 2^{\aleph_0}

. Let's not deny that.

John Baez (Dec 27 2020 at 23:34):

But the way you get these models may shed new light on what's "really going on" with these cardinals between

\aleph_0

and

c

Refurio Anachro (Dec 27 2020 at 23:48):

So the smaller models (like Gödel's universe) thinks that

c=2^{\aleph_0}

but in Cohen's models that cardinal is smaller than what they see as

c

You might want to read Scott's post starting from that quote, and you'll see what I mean.

Dan Doel (Dec 28 2020 at 00:08):

Is it actually known that the continuum hypothesis must hold in countable models? Those two things aren't obviously related to me.

Dan Doel (Dec 28 2020 at 00:10):

Dan Doel (Dec 28 2020 at 00:11):

Be sure not to conflate what appears true internal to the model vs. what appears externally true about the model.

Todd Trimble (Dec 28 2020 at 00:11):

Shooting from the hip: wouldn't downward Lowenheim-Skolem say 'no'? As always, one has to be careful around internal-external distinctions, but in a conversation about these things, this should go without saying.

Dan Doel (Dec 28 2020 at 00:12):

Todd Trimble (Dec 28 2020 at 00:19):

I admit I'm having trouble understanding what the original question is. I guess that the second cardinal in Refurio's OP is a Beth cardinal? (I don't read Hebrew.) Is it just

2^{\aleph_0}

? Refurio, is the goal to tease out an explanation of topos-theoretic forcing here on this Zulip? Or how far have you drilled down into this topic?

Refurio Anachro (Dec 28 2020 at 00:20):

That's not necessary, but I think it can look countable in another Cohen extended model (which, of course, would have its own c itself sees as uncountable).

Todd Trimble (Dec 28 2020 at 00:20):

Todd Trimble (Dec 28 2020 at 00:21):

Dan Doel (Dec 28 2020 at 00:21):

Something like Cantor's diagonal argument shows that internally

c

is uncountable. It may be modeled by a set that is countable according to the metatheory.

Refurio Anachro (Dec 28 2020 at 00:22):

Refurio Anachro (Dec 28 2020 at 00:23):

Todd Trimble (Dec 28 2020 at 00:26):

Yes. I still don't really know what sort of answer it is that you're looking for, i.e., what will be satisfactory to you. In order to find out, we might have to get into matters of your background. (Starting from ground zero in this zulip could be quite a long haul.) What have you tried reading so far?

Refurio Anachro (Dec 28 2020 at 00:37):

I'm not sure what I want to "tease out". I think I'm looking for a pair of models where the smaller inner

c

is some easy to understand cardinal.

\aleph_{\omega_{CK}}

might be easy enough, see my wild guess above. But other ones might do just as well.

I'd be happy to do some reading myself, and we really shouldn't need to start from zero. I wouldn't want to strain anybody's generosity. So if you see yourself able to give me some hints I'd be glad to work on them. Heck, knowing that I'm not asking for something impossible or silly would already be great!

Refurio Anachro (Dec 28 2020 at 00:38):

I got myself a copy of Moerdijk and MacLane's book 'Sheaves in Geometry an Logic', but haven't started reading yet. It looks like fun.

Dan Doel (Dec 28 2020 at 00:56):

What does "smaller inner

c

" mean? Does it mean that the set that models the internal

c

looks bigger externally? That is also probably unrelated to CH-type statements inside the model.

Todd Trimble (Dec 28 2020 at 01:19):

Mac Lane & Moerdijk is a really valiant effort that tries to steer a middle course between accessible pedagogy and getting the reader close to professional-level understanding. So I do think that is not a bad place to start. Consider also something like Kunen's book on set theory, which is more traditional but covers lots of ground in not too many pages (I've dipped into it, and found him helpful). Anyway you cut it, there will be some technical demands. There could also be some unlearning involved -- it sounds like you want a super-constructive approach. I can say that aleph sub church-kleene ordinal is not going to work, because of internal cofinality restraints. The truly original insights of Cohen must not be underestimated!

Dan Doel (Dec 28 2020 at 01:22):

I don't really understand that answer, either. Does it mean that you can't build a forcing model of

c = \aleph_{ω^{CK}_1}

due to the details?

Todd Trimble (Dec 28 2020 at 01:25):

Oh (Dan), you wrote

\aleph_{\omega_1^{CK}}

. I was referring to a countable ordinal, of countable cofinality. I'm sure you realize that

c

cannot have (internally) countable cofinality.

Todd Trimble (Dec 28 2020 at 01:28):

(There's something called König's lemma that's relevant here. There are two called by that name, but only one is relevant here.)

Dan Doel (Dec 28 2020 at 01:34):

You mean, more like

c = |ω^{CK}_1|

? Yeah, I would expect that to not work. I guess I was confused because the

\aleph_{...}

was already mentioned.

Todd Trimble (Dec 28 2020 at 01:44):

I was copying down exactly what Refurio wrote, which was

\aleph_{\omega^{CK}}

, where the subscript is the (countable) Church-Kleene ordinal. I was saying this could not be a possibility, because of König's lemma. I'm afraid that I don't know what

\omega_1^{CK}

signifies, I was just quoting that to say that's not what I was referring to.

Dan Doel (Dec 28 2020 at 01:44):

Todd Trimble (Dec 28 2020 at 01:45):

Dan Doel (Dec 28 2020 at 01:46):

Oh, well, in that case, I don't know enough about forcing aleph numbers to know that won't work. :smile:

Todd Trimble (Dec 28 2020 at 01:50):

Okay. The nLab article on König's lemma (written by me) might be less easy to follow than other standard sources. But it's a nice fact to know. I do recommend to anyone's attention the beautiful Easton's theorem, which puts very precise constraints on what is possible. But the proof is pretty advanced forcing theory. I think David Roberts (who occasionally tunes in here) is knowledgeable.

John Baez (Dec 28 2020 at 01:58):

Here I'll saying something that Todd already said... but I'll say it slower and louder:

There are countable models of every theory in first-order logic that you can write down with countably many symbols. This is the Lowenheim-Skolem theorem.

We can't really name uncountably many different things with just finitely many sentences written down using countably many symbols. So it turns out that whatever properties we say we want a mathematical structure to have, there's there's a countable structure with all those properties (if there's any structure with all those properties.)

Todd Trimble (Dec 28 2020 at 01:59):

John Baez (Dec 28 2020 at 02:00):

No, they're completely unrelated. There are countable models where CH holds, and countable models where it doesn't. Whatever finite collection of axioms you list (in a language with countably many symbols), if it has any models at all it has a countable model.

John Baez (Dec 28 2020 at 02:04):

@Todd Trimble - if you want to understand what Refurio is wondering about, you could look at Scott Aaaronson's post about the continuum hypothesis:

This is Scott clearing his throat before he tries to explain forcing. (He hasn't gotten around to doing that yet.)

John Baez (Dec 28 2020 at 02:06):

Well, actually you might have to read the 364 (!) comments on this post, where people try to lure Scott into going into more detail, and he valiantly tries to answer all his questions. You probably can't stomach that...

John Baez (Dec 28 2020 at 02:07):

Dan Doel (Dec 28 2020 at 02:09):

I think I have read the page on Easton's theorem, although it's not something I've retained.

I guess what I'm missing is assorted facts about cardinals, though (because I don't really study set theory). Does

ω^{CK}

having countable cofinality mean that

\aleph_{ω^{CK}}

also has countable cofinality?

John Baez (Dec 28 2020 at 02:09):

I don't actually think Refurio should read Sheaves in Geometry and Logic if he's trying to understand what models of

\text{ZF} + \neg\text{CH}

are like. It would first require a couple years of thought about category theory.

John Baez (Dec 28 2020 at 02:11):

I actually think the fastest way for Refurio to understand more about models of

\text{ZF} + \neg\text{CH}

is to either:

Dan Doel (Dec 28 2020 at 02:11):

John Baez (Dec 28 2020 at 02:13):

I think as long as

\beta

is a limit ordinal,

\{\aleph_\alpha \vert \alpha \lt \beta \}

is cofinal in

\aleph_\beta

John Baez (Dec 28 2020 at 02:14):

In other words, I think the function

\alpha \mapsto \aleph_\alpha

is "continuous".

Todd Trimble (Dec 28 2020 at 02:18):

John Baez (Dec 28 2020 at 02:21):

He's not a category theorist; I've mainly met him on G+ and Twitter talking about various other kinds of math.

Todd Trimble (Dec 28 2020 at 02:29):

Okay. Math is hard. :upside_down: I don't know any royal road to forcing, but at least some of this thread points to what is not possible, and already this is valuable knowledge.

David Michael Roberts (Dec 28 2020 at 03:16):

David Michael Roberts (Dec 28 2020 at 03:17):

Refurio Anachro (Dec 28 2020 at 10:54):

John is right, I am no expert on category theory, even tough I have stared at the subject for a long time now, and I believe I know some of the basics pretty well. I'm quite curious about sheaves and topoi, so I will give the according literature a try, no matter if it helps my current adventure or not. And yes, Scott Aaronson has been "clearing his throat", dropping lots of quite possibly unrelated stuff, hence I wouldn't expect anyone to read his blog post completely, let alone those 350+ comments, just to follow my thoughts!

Refurio Anachro (Dec 28 2020 at 10:54):

I'm also aware that he promised to continue this series, but looking at it I'm not convinced that he will manage to do so in the next couple of months, that's why I decided to attempt making my own way towards understanding forcing. And I think I had some success, but I'm very frustrated that what I've learned about his approach does not seem to tell us much about the meaning of those "cardinals in between". It should be easy to present one such cardinal, describe how it comes about, and what it means, right? That is what I'm trying to do!

Refurio Anachro (Dec 28 2020 at 10:57):

For the record, I'm not directly afraid of technicalities, but I may not be able to invest a lot of time to build a larger body of background knowledge, because I'm doing all this in my free time. In my experience, quite often the ideas and concepts can be explained in prose (or actually need to be, but that is often not written down), but, as we see in this thread, clarity may suffer if I don't know how to use jargon properly. I'm sorry for that. I'm trying.

Refurio Anachro (Dec 28 2020 at 10:58):

The result by Löwenheim-Skolem, up- and downward, is known to me. I couldn't (yet) follow their proof completely, though. I have errorneously used CK as a subscript for the Church-Kleene ordinal, and it's conventionally written

\omega_1^{CK}

. Also, writing things like

|\omega_1^{CK}|

seems like a very good idea. Thanks for that!

Refurio Anachro (Dec 28 2020 at 11:01):

I've read and I believe I understand König's paper. I also sensed that it applies to the Church-Kleene ordinal (either as index to

\aleph

, or as relative cardinality

|\omega_1^{CK}|

), but I am still confused about the relationship between models, and what they then think various cardinalities are, hence it appeared in my favorite wild guess. That is intentional to emphasize my confusion: If the effective topos cannot count beyond

|\omega_1^{CK}|

, is that then still a cardinal with countable cofinality (am I using cofinal correctly? I mean it's a supremum of a countable number of sets in the sense of König's theorem)?

Refurio Anachro (Dec 28 2020 at 11:03):

I can see now that

\aleph_{\omega_1^{CK}}

can never be the cardinality of the continuum, but maybe it still makes sense to say something like

c=|\omega_1^{CK}|

Refurio Anachro (Dec 28 2020 at 11:04):

Cohen's forcing method to prove the independence of CH starts with some model

A

, which has an idea of what it's cardinality of the continuum

c_A

is, and then adds new reals to it, establishing a new model

B

which has its own idea of the cardinality of the continuum

c_B

. Due to the extension method I assumed that

B

contains

A

, and that

B

can look at

c_A

to compare it with it's own

c_B

. I hope that this clears up what I meant by "inside" and "outside" (of

A

Refurio Anachro (Dec 28 2020 at 11:06):

The reason I started talking about countable models is that it seems to me that in the "outside view" of a countable model

A

, that is, from a model

B

where we can see

A

as countable, we should also be able to see that

c_A

is countable, right?

Refurio Anachro (Dec 28 2020 at 11:07):

I'm well aware that we can only ever say countably many things (in a somewhat unrelated fact even only finitely many) about any model of set theory, and that that is related to what we mean by countable model. I'm not an expert on this, but it seems that topos theory gives a nice avenue to make this precise. Maybe I should think about "smaller" models (in the

A

B

sense above) than the effective topos.

Refurio Anachro (Dec 28 2020 at 11:11):

In any case, I am thankful for your discussions in this thread! It does give me a couple of starting points for continuing my quest, which should keep me busy for a while.

Also, hi @David Michael Roberts, nice to see you here! Yes, as you say, I should have written

\aleph_1=2^{\aleph_0}

, and I should have known better. Thanks!

Refurio Anachro (Dec 28 2020 at 11:23):

Re @John Baez: I did ask Scott on his blog just yesterday. I'm not sure if I'm happy with the way it came out, as I feel I could do much better now.

I am not quite convinced that it's a better place for this discussion, as he may not yet be ready to answer my question, and because he does not seem to be into topos theory, which I'd be happy to involve. Also, the crowd in the comment section over there may not compare with this terriffic forum of experts here.

In any case, as I mentioned earlier, I am just an amateur, so it's important to me that you nice people see some value in this thread, or even have fun pondering my question. I really wouldn't expect answers on a silver platter. Enjoy yourselves and Merry Christmas!

Todd Trimble (Dec 28 2020 at 11:28):

|\alpha|

denotes the cardinality of an ordinal

\alpha

, then of course not, because

\omega_1^{CK}

is countable.

Refurio Anachro (Dec 28 2020 at 11:32):

Nice hint, @Todd Trimble! It seems, I really don't understand what "countable model of ZFC" means. I'll see if I can look that up after having breakfast.

Todd Trimble (Dec 28 2020 at 15:00):

Well, it means countable from the point of view of the "meta-background". As near as I can tell, the usual maneuver of the traditional set theorist is to start with a countable transitive model M of ZFC and then produce a forcing extension M[G], all these constructions being small so that these are actual objects in the universe V that is the "real world" of sets that forms the backdrop. For a category theorist used to dealing with large categories, this feels sort of fussy and uptight. My gut inclination would be to say: don't worry much about countable models; they're a crutch.

John Baez (Dec 28 2020 at 17:20):

To talk about a model of set theory you need to already be working within set theory, so we start with our universe

V

of sets and the usual membership relation

\in

; let's assume it obeys ZFC.

A model of set theory is then a collection

U

of sets where

U \subseteq V

and a relation called

\varepsilon

that we use to mean membership in the model even though it may not be the real

\in

John Baez (Dec 28 2020 at 17:21):

We require that the sets in

U

and the relation

\varepsilon

obey the ZF axioms (let's use those axioms).

John Baez (Dec 28 2020 at 17:22):

So for example there must be an "empty set in the model". That's a set

E \in U

such that for no set

X \in U

do we have

X \varepsilon E

John Baez (Dec 28 2020 at 17:23):

John Baez (Dec 28 2020 at 17:24):

John Baez (Dec 28 2020 at 17:26):

Okay, so that's a model of set theory in a nutshell! Ask questions if it's unclear!

John Baez (Dec 28 2020 at 17:27):

Then it's really easy to say what's a countable model of set theory. It's a model

(U,\varepsilon)

such that

U

is countable!

Dan Doel (Dec 28 2020 at 17:41):

Yeah, it seems like a main problem you're having is mixing 'internal' and 'external' things together, and it's really important to not do that.

For instance, I think you mentioned the effective topos earlier. There the model is build so that everything is externally paired up with some method of computing it. Since there are only countably many 'programs', every set used in the model is externally countable. However, the diagonal argument is constructive, and can be used to show that e.g.

2^ℕ

is uncountable internally.

When you decode what the internal statement means externally, it is that the computable functions from

ℕ

2

are "computably uncountable," meaning that for every computable enumeration

ℕ → 2^ℕ

, we can compute an element of

2^ℕ

that is not in the enumeration.

Dan Doel (Dec 28 2020 at 17:42):

And of course, even though the external set theory thinks the set used for

2^ℕ

in the model is countable, the bijection exhibiting that is not computable.

David Michael Roberts (Dec 29 2020 at 05:39):

Absolutely. It's just so that the Rasiowa–Sikorski lemma can be applied so that a generic filter can be said to exist, which is equivalent to what is needed to pass from the category of double negation sheaves, via the filterquotient construction, to a well-pointed topos. If you don't care about that, and are happy with either boolean-valued models or internal logic, then countability is absolutely not needed.

Fawzi Hreiki (Dec 29 2020 at 12:11):

Well if you want to prove independence from ETCS then you need well-pointedness no?

Refurio Anachro (Dec 29 2020 at 13:11):

Thanks, @John Baez, this is indeed helpful. I'm still not quite convinced that I won't want to think about countability issues, but I'll take your suggestion into account @Todd Trimble. Thanks, @David Michael Roberts, I was just about to complain when I read your extra breadcrumbs. It made me reconsider what I might want with countability, and I'm now more confident that I won't have to look at this to follow you guys. Special thanks to @Dan Doel for keeping an eye on me.

Refurio Anachro (Dec 29 2020 at 13:12):

Looking at what wikipedia has to say about forcing I slowly realize that Cohen's original 'inside' view is nowadays considered more difficult than an approach based off a backdrop set theory. Especially Dana Scott's boolean valued models. Well, yes, that makes my questions seem... a bit ill-informed. I still have the urge to look at this in a completely relative way, without any backdrop theory. Curious what MacLane and Moerdijk have to say about this. Hope I'll find the time to learn what's fussy and uptight about the set theory way of doing stuff.

Refurio Anachro (Dec 29 2020 at 13:15):

Forcing seems much simpler now, and I think I should be able to construct an example like I've been asking for, even if that seems much more artificial than I had suspected earlier XD

Todd Trimble (Dec 29 2020 at 13:48):

Thank you! I'll know whom to turn to if I have questions about forcing. When I first tried learning about forcing (the traditional way), I was mystified by the need to introduce Martin's axiom, but from the Wikipedia article, it seems to be to take care of the existence of the generic filter for uncountable models.

Todd Trimble (Dec 30 2020 at 03:10):

@David Michael Roberts But I have to say, the need to invoke things like Martin's axiom (which is independent of ZFC if I recall correctly) to construct generic filters in the case of uncountable models seems like a very poor advertisement for the traditional set-theoretic approach to forcing. In contrast, the topos-theoretic approach -- and let's say we satisfy the demand for two-valued models by passing to a filterquotient in the last step -- doesn't seem to involve any of this Martin's axiom stuff. So it seems that the topos-theoretic approach requires fewer extraneous assumptions! Is that right? If not, could you set me straight on this?

David Michael Roberts (Dec 30 2020 at 03:45):

Hmm, to construct the filterquotient you still need some input, but maybe it's less than what the material version needs.

Todd Trimble (Dec 30 2020 at 04:21):

I don't see anything too technical. There is Proposition VI.1.6 from Mac Lane-Moerdijk which reads: let E be a Boolean topos and let U be a maximal filter of subobjects of 1 in E. Then the filter-quotient topos E/U is two-valued (and again Boolean). There is no indication that anything more is needed.

André Beuckelmann (Dec 30 2020 at 11:54):

As far as I'm aware, you won't really need Martin's axiom, if you are working with countable models - Martin's axiom is needed to prove the existence of a filter intersecting a family of dense sets of cardinallity strictly between

\aleph_0

and

2^{\aleph_0}

, but, as your model is countable, it doesn't know any such family. Also, given that every model of ZFC has a forcing extension satisfying Martin's Axiom, I don't think there is any harm in assuming that the model you are starting with satisfies it. The advantage of Martin's axiom, on the other hand, is that its independence allows you to prove the independence (or, rather, consistency) of many other statements - it can, in some sense, be interpreted as saying that, in the absence of CH, the cardinals between

\aleph_0

and

2^{\aleph_0}

are almost as well-behaved as

\aleph_0

itself - otherwise, they can be incredibly wild - so it has a number of useful consequences.

Generally, as far as I understand, the topos theoretic way of doing forcing is quite nice for simple forcing, but we haven't yet figured out a good way to do iterated forcing with it, which is needed, sometimes (for example, to establish the consistency of

\text{ZFC}+\neg \text{CH}+\text{MA}

. Also, as ZFC is not finitely axiomatisable, I don't think there is any nice characterisation of topoi that correspond to models of ZFC - one usually only gets a weaker set theory.

Todd Trimble (Dec 30 2020 at 13:07):

I thought I'd heard David Roberts had done some work on iterated forcing via topos theory, but obviously he should be the one to say. (Scedrov too? I don't recall exactly.)

There are ways to take into account replacement schema and so forth for toposes. Mike Shulman has developed much of this in the context of his stack semantics.

Todd Trimble (Dec 30 2020 at 13:21):

Forgot to ask: what do people do with

\text{ZFC} + \neg\text{CH} + \neg\text{MA}

André Beuckelmann (Dec 30 2020 at 14:13):

@Todd Trimble Do you mean

\text{ZFC}+\neg\text{CH}+\text{MA}

\text{ZFC}+\neg\text{CH}+\neg\text{MA}

? For the latter, I don't know if there is any good reason for it. The former, for example, proves that there is no Suslin line (i.e. it proves that the real numbers are the only complete, dense linear order without end points that has the countable chain conditions).

Todd Trimble (Dec 30 2020 at 14:14):

André Beuckelmann (Dec 30 2020 at 14:24):

I'd guess so? It is provable that the analogue of Martin's axiom is false for

2^{\aleph_0}

, so you'd probably want to force

2^{\aleph_0}

to be larger in a way that doesn't create too many new filters. Perhaps even the normal Cohen forcing might already do it.

John Baez (Dec 30 2020 at 16:11):

I don't know anything about any of this, but the continuum hypothesis implies Martin's axiom, right? So

\text{ZFC} + \neg\text{CH} + \neg\text{MA}

is the same as

\text{ZFC} + \neg\text{MA}

, right?

André Beuckelmann (Dec 30 2020 at 16:23):

Todd Trimble (Dec 30 2020 at 16:24):

John Baez (Dec 30 2020 at 17:08):

I'd never looked into Martin's axiom until just now, but it's a funny axiom, since it says all those mysterious cardinals between

\aleph_0

and

2^{\aleph_0}

behave like

\aleph_0

in some respects. Sort of like an axiom saying that all the ghosts in my house eat the same thing for breakfast as me and my wife.

Todd Trimble (Dec 30 2020 at 17:37):

Yes. So to put your earlier remark in those terms: that's of course true if there are no ghosts in your house. :-)

Just now I'm reminded of a peculiar fact I learned some time back: ZF + GCH implies the axiom of choice. I don't exactly know why I'm bringing this up; just some of the fun and games to be had with Hartogs numbers.

John Baez (Dec 30 2020 at 19:52):

Yes, I believe Martin's axiom, and I believe all the ghosts in my house eat oatmeal for breakfast just like Lisa and me.

David Michael Roberts (Dec 30 2020 at 23:10):

Definitely Ščedrov, I just picked up that one can do many instances of proper class forcing this way, too (and despite one claim I made, not all instances)

David Michael Roberts (Dec 30 2020 at 23:14):

Thanks for checking, @Todd Trimble . I'm on holidays without the usual resources to hand and was not in a position to dig them out. At one point I thought about what the 'generic' is, in the setting of material set theory, from the point of view of the double-negation sheaves, but I've forgotten,

Valeria de Paiva (Dec 31 2020 at 01:56):

Valeria de Paiva (Dec 31 2020 at 03:11):

John Baez (Dec 31 2020 at 05:03):

John Baez (Dec 31 2020 at 16:10):

By the way - I never said that. But I could have said it, because I would never want to deny that in models where CH fails, CH fails.

Refurio Anachro (Dec 31 2020 at 16:34):

Is Martin's axiom compatible with lots of space before the continuum, e.g.

c=\aleph_{\omega_1+1}

Valeria de Paiva (Dec 31 2020 at 16:37):

John, what do you mean you never said that? you did, I was only replying to a specific sentence of yours on the 27 Dec? just a bit earlier in this thread, I think. I sometimes spend 4/5 days without logging on to zulip and there's lots to read, but I was only replying to what I thought was the implied question: which are these middle cardinalities that exist if CH fails? JohnBaez27Dec.png

Refurio Anachro (Dec 31 2020 at 17:20):

Heh, that's why I started this thread! Well, I made some weird typos, and I think one of them prompted John's statement you quoted. Which didn't stop me from doing it again %-D

Anyways, I read John's answer such that he did not say anything about Cichoń's diagram, which he could have (I know he's heard of it), but that that wouldn't have helped the confusion, of which I had plenty. Which can be contagious. Which it hadn't been in this case. Yet...

John Baez (Dec 31 2020 at 19:22):

Oh, wow, I did say that. Sorry. I looked back to see where I said it, but I didn't look back far enough - I was only examining my recent remarks. By the way, if you use the "quote" feature on Zulip your quote will contain a link to the original remark. Like this:

John Baez (Dec 31 2020 at 19:24):

By the way, I was not offended, just puzzled. I couldn't figure out why I would say this as part of the recent conversation about Martin's axiom.

Valeria de Paiva (Dec 31 2020 at 20:58):

yeah, I have not learned how to use quote features in any of the platforms. I thought I was using it.
here it doesn't bother me much. but on whatsapp, it's a problem.

John Baez (Dec 31 2020 at 21:01):

To quote a comment, add an emoticon to it, or do all sorts of other wonderful things, I just move my mouse to the right end of your comment and click on the little options that appear.

John Baez (Dec 31 2020 at 21:02):

It's possible you quoted my message but then deleted the stuff that creates a link back.

John Baez (Dec 31 2020 at 21:03):

@_**John Baez|275920** [said](https://categorytheory.zulipchat.com/#narrow/stream/266967-general.3A-mathematics/topic/Forcing.3A.20cardinals.20below.20c/near/221315962):

Nathanael Arkor (Dec 31 2020 at 21:11):

On my screen, Valeria's messages correctly quote yours, John. Perhaps there's some issue with Zulip.

John Baez (Dec 31 2020 at 21:13):

Refurio Anachro (Dec 31 2020 at 22:16):

André Beuckelmann (Dec 31 2020 at 23:14):

If inaccessible cardinals are consistent, you can even have

\text{MA}+2^{\aleph_0}\text{ is weakly inaccessible}

Todd Trimble (Dec 31 2020 at 23:39):

Refurio, if you haven't already, look up Easton's theorem. Prepare to have your mind blown. If your mind isn't blown, you might not have appreciated Easton's theorem. :-)

John Baez (Dec 31 2020 at 23:44):

Refurio Anachro (Jan 01 2021 at 00:23):

Which says that the only constraints on powersets are the theorems by Cantor (powersets are strictly larger), and König (it mustn't be the supremum of a countable set of cardinals). Under ZFC, as AC doesn't add any further constraints. Yes, that's rather extreme. I knew the fact, but not that it's due to Easton. Thanks for that, @Todd Trimble! And Solovay's earlier result says the same, but only for

c=2^{\aleph_0}

The thing with MA is that it's supposed to contain that mess somewhat. Which it... I suppose it does!!?! And it felt so reasonable, that it's effect would be to make all cardinals less than

c

kind of countable.

Todd Trimble (Jan 01 2021 at 00:34):

But it's not just one single infinite power set cardinality that can be manipulated according to these constraints. It's all of them, simultaneously. See the statement "if

G

is a class function..."

Refurio Anachro (Jan 01 2021 at 00:52):

So G just has to shift cardinalities! It could be larger than the powerset operation (say, the ε map), but it could also be smaller, since CH does not hold.

That's gross! ZFC is a joke! I'm starting to feel why Woodin might have changed his mind to argue for CH!

John Baez (Jan 01 2021 at 02:10):

I think one reasonable attitude is to treat "a model of ZFC" as a bit like "a model of the theory of groups". We don't expect the axioms for a group to uniquely pin down a specific group... so we're not shocked that there are lots of very different groups. Similarly there are lots of very different models of ZFC. The problem is that we were raised to talk about "the" universe of sets, not "a" universe of sets.

Todd Trimble (Jan 01 2021 at 02:25):

I don't think you said that right. Here

G

is supposed to be a possible power set operation. Easton says that one can force

G(\kappa) = 2^\kappa

to occur for all infinite cardinals

\kappa

provided that

G

satisfies these conditions:

\kappa \leq \lambda

implies

G(\kappa) \leq G(\lambda)

and

\kappa < \text{cof}(G(\kappa))

and

G(\kappa)

equals its own cofinality. (The first two conditions are necessary conditions. The last isn't, but it's what Easton assumed to prove his theorem. Weakenings of the last condition are not too well understood, but there has been some progress since Easton.)

What gets me is that you can have things like

2^{\aleph_0} = 2^{\aleph_{16}} = \aleph_{17}

: that the power set operations need not be injective! But of course it has to be monotone, and satisfy the cofinality condition mandated by König's theorem.

Refurio Anachro (Jan 01 2021 at 19:45):

So when

2^{\aleph_0}=\aleph_{17}

it does not contradict Cantor's theorem that

2^\kappa>\kappa

when we also have

2^{\aleph_1}=\aleph_{17}

, because we had skipped that. Nifty!

No, I had some strange ideas about other functions and I cannot remember what they might have been. It was late for me, and that Wikipedia entry is a bit spartanic, too.

Refurio Anachro (Jan 01 2021 at 19:50):

But I'd like to stick with my appallment, even if only for the sake of this discussion. ZFC introduces all those new objects, demanding a rather excessive ontology for them, and then pretty much completely fails to pin down their meaning with respect to the powerset operation, a basic tool of set theory! Admitted, that operation is what prompted Cantor to start introducing all those sets...

Todd Trimble (Jan 01 2021 at 20:02):

Well, that's a good remark: we have here a death blow to a type of Platonic dream that there's a truth of the matter "out there" and it's up to us to find it. Joel David Hamkins has been writing and theorizing about the set-theoretic multiverse, and I think topos theory also encourages the outlook that there's not a single hegemonic monolithic foundations, but a network of different possible worlds of mathematics.

Every time I hear various philosophers or foundationalists banging on about ontology and ontological commitment and so forth, I get a little nervous, as if someone is trying to sell me a set-theoretic bill of goods. For example, I hear various people trying to explain that ETCS has a "weak ontology". I struggle to understand exactly what they mean by that.

Refurio Anachro (Jan 01 2021 at 20:02):

I seem to have much less trouble accepting large ordinals, but I'm sure you could just as easily mention a breadcrumb that would nuke that conception of mine.

What was that somewhat popular alternative, weaker set theory again, where we don't need cardinals beyond

\aleph_2

. Was it Kripke-Platek set theory? I remember that KP's consistency strength is the Feferman-Schütte's ordinal

\Gamma_0

(I once wrote a post about this on g+), but that's still countable, so it might be. Or was it an even weaker theory? I tried to find a reference, but it's eluding me :S

John Baez (Jan 01 2021 at 20:04):

If you're appalled by the fact doesn't pin down the cardinality of

2^{\aleph}_\alpha

, just assume the generalized continuum hypothesis! It's sort of like: if you're appalled by nonabelian groups, assume your group is abelian. :upside_down:

Personally - since I'm not really a set theorist - I'd be happy to use ZF + V = L. A bunch of set theorists object to it:

But these objections seem unconvincing to me. I sort of like the idea of working in a universe where you have more of a handle on what's there, so it doesn't seem "unnecessarily restrictive" to assume the only subsets are the constructible ones.

John Baez (Jan 01 2021 at 20:07):

Maybe they're trying to say "There are more things in heaven and Earth, Horatio, than are dreamt of in your philosophy". :upside_down:

Refurio Anachro (Jan 01 2021 at 20:09):

Just for the record, I actually like Hamkin's approach a lot, and usually I'd feast on philosophers struggling with the ontology of set theory, especially when they've been ignorant of it }:->

But I also feel it's my duty to have a good intuition about these things, to earn my right to troll about that stuff. So I keep coming back to the platonic way of looking at it, even if ETCS seems very satisfying, especially from an operational (or computational) point of view.

Dan Doel (Jan 01 2021 at 20:14):

Refurio Anachro (Jan 01 2021 at 20:16):

I think V=L is lacking some cardinals useful in model theory, it limits our ability to measure consistency strength in some important or interesting cases. Model theorists seem to love GCH, though.

Dan Doel (Jan 01 2021 at 20:16):

Also, every proof theoretic ordinal is countable (and less than

ω^{CK}_1

, I think).

Todd Trimble (Jan 01 2021 at 20:20):

I have a feeling they're trying to say something else. Something along the lines that ZFC in conjunction with the idea of the cumulative hierarchy gives a satisfying intuition for what sets are, and that ETCS looks more like just some jumble of axioms without a coherent underlying intuition for what's behind them. There's an idea that the cumulative hierarchy is recursively germinated from something small: for example, the hereditarily finite sets germinate just from the empty set. It's that recursive generation which imparts a feeling of a robust ontology. Or something like that.

Refurio Anachro (Jan 01 2021 at 20:21):

That seems meaningful, Dan Doel. I suppose I got that backwards, then. These larger things aren't needed to measure things, but to construct them? Sorry, I keep ending up fumbling.

Todd Trimble (Jan 01 2021 at 20:22):

I don't think you have to worry about fumbling, Refurio. We're just chatting informally. At least, I hope we are.

John Baez (Jan 01 2021 at 20:25):

I don't think "platonism" per se gives one much intuition about set theory. I think different set theorists have different intuitions about what sort of universe of sets they'd like - or phrased more platonically, what the universe of sets "is like". It would take a lot of work to get these intuitions, and it would be really interesting. I just find it tiresome when the conversation is phrased in terms of which intuition is "correct", in cases where two different intuitions lead to two different theories that could both be consistent.

There's one intuition that seems to say "anything not forbidden should be allowed". This pushes for bigger and bigger large cardinals, etc. That's certainly fun to explore.

Dan Doel (Jan 01 2021 at 20:27):

2^{\aleph_0} = 2^{\aleph_{16}} = \aleph_{17}

kind of doesn't seem that weird to me, but then I'm used to cardinalities not being a total order. :smile:

Dan Doel (Jan 01 2021 at 20:28):

Refurio Anachro (Jan 01 2021 at 21:02):

Platonism is just an excuse to evoke a peculiar kind of existential angst to get in touch, with set theory concepts in particular, in a performance art kind of meditation move, @John Baez. It's just a trick. If someone would press me to choose, I'd go for the larger universe. I'm not sure if that'd be the most helpful thing to do, as a backdrop to better understand what we're doing, or to explore the boundaries of our feeble not-even-countable understanding of maths.

Refurio Anachro (Jan 01 2021 at 21:04):

John Baez (Jan 01 2021 at 21:16):

I'd like to know enough about set theory to appreciate the intuitions of the experts... like why Woodin wants the axiom "V = ultimate L":

I doubt I could ever be convinced that these intuitions are "true" in some absolute sense, but perhaps someday I could understand why they're appealing.

Refurio Anachro (Jan 01 2021 at 21:41):

In what sense could some systems be more true than others? Contradictions are clearly bad. Systems too small to contain popular, or interesting maths would surely be out, too. But what counts as interesting? Gödel's incompleteness result suggests, that we'd never be able to find a formalism to encompass everything. At least as long as we only consider consistent or countably generated systems...

Yes, I'd be curious to learn about such intuitions, too! I have occasionally tried to follow Woodin, but only with very limited success. Then there is this conundrum of identifying the standard counting numbers. Maybe that's less mysterious than I think, but I'm certainly not done digesting Nelson's work about this. People must have different approaches and values, right?

John Baez (Jan 01 2021 at 22:55):

I don't know! But people like Woodin seem to have some intuitions about set theory, going beyond the axioms, that make them feel some axioms are "true" and others not.

It has a lot of nice history, and isolates some of the intuitions that set theorists use to say what's "true". For example:

John Baez (Jan 01 2021 at 22:56):

John Baez (Jan 01 2021 at 23:01):

Yeah, I don't understand his work at all. It's pretty low-priority for me, too. It just intrigues me that there's a guy like this who seems to have some intuition about the universe of sets that motivates him to prove lots of big theorems....

David Michael Roberts (Jan 02 2021 at 05:12):

As far as I recall, Hamkins' (and others) look at the multiverse from the point of view of set forcing models, but not (yet) from class forcing models. Happy to be corrected.

David Michael Roberts (Jan 02 2021 at 05:15):

I'd be careful about saying that the cardinality of every power set can be manipulated, because I think the power set of the natural numbers can be singular, just not with countable cofinality. And Easton's theorem only says that regular cardinals can be adjusted as desired, subject to the two mild constraints coming from Cantor and König. The singular cardinals are much weirder and harder to understand.

Todd Trimble (Jan 02 2021 at 16:50):

@David Michael Roberts Oh, you're right that I misread the statement from the Wikipedia article. This is embarrassing, as I'm pretty sure I'd gotten this right in the past.

Stream: deprecated: mathematics

Topic: Forcing: cardinals below c

Refurio Anachro (Dec 26 2020 at 22:28):

John Baez (Dec 27 2020 at 01:39):

Matteo Capucci (he/him) (Dec 27 2020 at 08:51):

Matteo Capucci (he/him) (Dec 27 2020 at 08:52):

Refurio Anachro (Dec 27 2020 at 12:41):

Refurio Anachro (Dec 27 2020 at 12:43):

Matteo Capucci (he/him) (Dec 27 2020 at 13:04):

Matteo Capucci (he/him) (Dec 27 2020 at 13:05):

Matteo Capucci (he/him) (Dec 27 2020 at 13:06):

Matteo Capucci (he/him) (Dec 27 2020 at 13:06):

Refurio Anachro (Dec 27 2020 at 14:14):

Todd Trimble (Dec 27 2020 at 15:24):

Matteo Capucci (he/him) (Dec 27 2020 at 15:46):

Matteo Capucci (he/him) (Dec 27 2020 at 15:48):

Matteo Capucci (he/him) (Dec 27 2020 at 15:49):

John Baez (Dec 27 2020 at 16:06):

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Matteo Capucci (he/him) (Dec 27 2020 at 17:02):

John Baez (Dec 27 2020 at 17:41):

Fawzi Hreiki (Dec 27 2020 at 17:48):

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Dan Doel (Dec 27 2020 at 18:02):

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Dan Doel (Dec 27 2020 at 18:15):

Dan Doel (Dec 27 2020 at 18:18):

Matteo Capucci (he/him) (Dec 27 2020 at 18:29):

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Dan Doel (Dec 28 2020 at 00:08):

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Dan Doel (Dec 28 2020 at 00:11):

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Refurio Anachro (Dec 28 2020 at 00:37):

Refurio Anachro (Dec 28 2020 at 00:38):

Dan Doel (Dec 28 2020 at 00:56):

Todd Trimble (Dec 28 2020 at 01:19):

Dan Doel (Dec 28 2020 at 01:22):

Todd Trimble (Dec 28 2020 at 01:25):

Todd Trimble (Dec 28 2020 at 01:28):

Dan Doel (Dec 28 2020 at 01:34):

Todd Trimble (Dec 28 2020 at 01:44):

Dan Doel (Dec 28 2020 at 01:44):

Todd Trimble (Dec 28 2020 at 01:45):

Dan Doel (Dec 28 2020 at 01:46):

Todd Trimble (Dec 28 2020 at 01:50):

John Baez (Dec 28 2020 at 01:58):