## Large Cardinals and their Justifications

5 12 2005

Most mathematicians are willing to use ZFC (or something fairly similar) as a foundation for their work. However, since Gödel, we know that these axioms are essentially incomplete – that is, any consistent, recursive extension of them will still leave various statements undecided. Some people seem willing to say that this just means that there is no fact of the matter about statements that go beyond ZFC, but I think this is just too hasty.

After all, if we are platonists, then we think that ZFC is true, and thus consistent. And if we are fictionalists, we at least think that it’s a good story, but part of being a good story seems to require consistency. In fact, on just about any reasonable view of mathematics, it seems that however much we can actually say to be the case should at least be consistent. Thus, whatever theory T we use, we should probably be willing to say “T is consistent” as well.

Now, “T is consistent” is not itself a mathematical statement, but there is a susbtitute, which I will call “Con(T)”, which states that no natural number has a certain property, which we intuitively understand as saying that it codes a proof from T of a contradiction. Whatever we might or might not be able to say about the natural numbers, I think we understand them well enough to say that this proof-coding mechanism is actually correct. (You can think of this coding as the way that a computer codes text, and can reason syntactically about this text to see which strings are correct proofs in our formal system – there’s another coding that’s easier to work with and is therefore more standard in the literature, involving powers of primes.) Thus, if T is an appropriate formalization of some part of the truths (or useful fictions, or whatever) of mathematics, then Con(T) should be as well.

Now, by Gödel’s Completeness theorem (which requires some fragment of ZFC to prove), we know that if Con(T) is the case, then there is actually some set and a collection of relations and functions on that set that can be used as the interpretation of the symbols in T to make it come out true. That is, any consistent theory (according to the numerical coding) has a model. Thus, if we think that ZFC+Con(ZFC) is a reasonable formulation of some part of mathematics, then there must in addition be some model of ZFC. If in addition, we assume Con(ZFC+Con(ZFC)), then there is a model of ZFC, which itself contains a model of ZFC.

Now, these models of ZFC may bear very little resemblance to the “real thing”. In particular, the symbol for the set membership relation may be interpreted as some relation that has no connection to the real thing. In fact, there is a model whose domain is just the natural numbers, and whose “set membership relation” is some relation among numbers. But if there is an inaccessible cardinal I, then there is in fact a set (called VI) such that using the actual set membership relation on this domain creates a model of ZFC. This set contains all members of its elements, so it actually “knows” what sets they are (unlike one of those models whose “sets” are natural numbers, and assigns them elements in some seemingly arbitrary way), and also contains the actual powersets of its elements (and thus all their subsets, not just some of them), and thus has a lot more properties in common with the whole of mathematical reality than the other sorts of models of ZFC. In fact, it contains all the natural numbers and knows which set is the set of all natural numbers, so it knows any true statement (or true according to the fiction of mathematics, or whatever) of the form Con(T). So if there is an inaccessible cardinal, then VI is a model of ZFC, so Con(ZFC) is true, so VI is also a model of ZFC+Con(ZFC), so Con(ZFC+Con(ZFC)) is true, so VI is also a model of Con(ZFC+Con(ZFC+Con(ZFC))), etc. So finding a model of this appropriate sort guarantees a lot of consistency statements.

As I said above, we have good reason to adopt all these statements. In addition, for any (even non-recursive) combination of consistency statements, a model of the form Vk satisfies all of them, so in some sense, saying that a model of this form exists just says that these non-recursive sets of axioms are consistent, just as the statement Con(T) said that a certain recursive set of axioms was consistent. Thus, we seem to have good reason to adopt the existence of an inaccessible as well as all the consistency statements.

Of course, once we have adopted ZFC+Inaccessible, we can start the whole process all over again, and eventually reach ZFC+”There are at least two inaccessibles”. Similar arguments get us arbitrarily large numbers of inaccessibles (even into infinite orderings of them).

I’m told that larger large cardinal axioms state similar properties about the universe and guarantee that we can bring them down to models of a certain form, extending Gödel’s Completeness Theorem.

Now, I’ll consider the endpoint of all this. The collection of all true (or fictional, or whatever) statements of set theory should be consistent. Thus, to extend Gödel’s Completeness Theorem, there should be some model of the form Vk that satisfies all true statements of set theory, ie, is elementarily equivalent to the universe as a whole. Now, the statement that such a model exists is certainly not consistent if it can be stated (because it would violate Gödel’s Second Incompleteness Theorem, saying that no theory can assert its own consistency, or equivalently the existence of a model of the entire theory). Perhaps more relevantly, I think it can’t even be stated, because Tarski showed that we can’t define the truth-predicate for a model inside that very model, so we can’t state that a model of “all true statements” exists. So this statement can’t be adopted as a “final large cardinal”.

However, even if it could be stated, or if it were consistent, we might still want to go farther. Since the universe contains an elementary submodel of this form, let us say, Va, then Va must also contain such a model, say Vb. Thus, there should be a and b such that Vb is an elementary submodel of Va. And this statement can in fact be expressed in the language of set theory.

I don’t know if this statement can be shown to be inconsistent (which would call into question this means of justifying large cardinals). It might also be equivalent to some already-known large cardinal axiom (my friend Adam Booth suggested to me that it sounds like it could be a consequence of the existence of an inaccessible that is the limit of a set of measurables). If anyone with a relevant background in set theory could tell me, that would be great.

Anyway, all this is just one means of justifying large cardinal axioms, but it seems to make sense to me. It also has the added benefit of not requiring a platonist view of mathematics, but works also on a fictionalist view, and probably a variety of other views as well. All that is needed is to assume that whatever set of axioms we should adopt should be consistent, and that the set of natural numbers that our axioms describe should satisfy Con(T) iff T is actually consistent. Of course, there will still be further statements that we won’t be able to decide with a principle like this, but it gives us a means of going far beyond ZFC. Recent work of Hugh Woodin and others suggests that it won’t be enough to settle the Continuum Hypothesis however.

### 14 responses

6 12 2005

Hi Kenny–

Another enlightening post — your basic idea strikes me as intuitive and attractive. Your justification also seems to me to be ‘internal’ (in the sense of Carnap’s internal vs. external question dichotomy). In other words, it seems like something I might hear from the professor in a class on set theory — on the day the large cardinal axioms are introduced, your story would be the rationale run through to justify their introduction. (As opposed to something you might hear down the hall in the philosophy of mathematics seminar.)

One question, born of mathematical ignorance: are any important (in the sense of useful, fruitful, or whatever) large cardinal axioms left out before we reach the stage described in your post under “Now for the endpoint of all of this”? That is, I’m wondering whether you could just end your story before that paragraph begins, leaving out what follows, and still have provided a justification for all the larger cardinals that mathematicians really want/ need/ use.

6 12 2005

This is where I need to learn more. I can see how this justification goes for inaccessibles, and I know something related for measurables, but it’s not as convincing. (We end up with an elementary embedding of the whole universe inside itself as a proper class, rather than an elementary equivalent set model as I recommended towards the end.) However, I don’t know how to justify Mahlos, hyper-Mahlos, or any of the other things between inaccessibles and measurables. And I don’t know enough about the definitions of cardinals above measurables to begin justifying them. Except that having countably many Woodin cardinals plus a measurable cardinal above all of them is equivalent (or equivalent in consistency strength?) to projective determinacy. I still haven’t managed to understand the statement of what a Woodin cardinal is yet. I’m hoping to learn more about some of this stuff next semester in Woodin’s seminar.

I’ve seen statements of Ramsey, Rowbottom, weakly compact, and maybe strongly compact cardinals, but only briefly, and I need to learn more about what sorts of embedding properties they apply to the universe. But I’m told that many of them generalize this sort of “reflection” principle of saying that certain submodels of ZFC reflect properties of the whole universe.

I’m more interested in the extrinsic justifications through things like projective determinacy, and I think those arguments can be made on non-platonistic accounts of mathematics too. But as you point out, that’s a different story.

9 12 2005

This doesn’t justify large cardinals at all! What you want is for your theory to model Con(ZFC) and Con(Con(ZFC)) etc.. These statements can be coded up just as number theoretic statements therefore any two models which agree on the natural numbers agree on these statements. Supposing ZFC is actually consistant all these statements will be true in *any* well-founded model of set theory! Since your arguments starts with the premise that our models are actually well founded, i.e., our coding mechanism is valid, the large cardinals get you nothing new.

Sure it is interesting that the large cardinals prove these number theoretic facts but infering large cardinals from this fact is clearly in error. You would never let your 12A students get away with arguing Y is true X->Y hence X is true so how do you get away with arguing that since large cardinals prove these Con statements you want to be true the large cardinals must be true as well. I mean the assumption their are only finetly many primes would help me prove a whole lot of true statements in number theory so is that grounds to suppose this statement is true?

Finally I would point out your very argument undercuts itself. Unless you are actually assuming platonism, which you seem to want to avoid, we can’t depend on the V_i’s actually reflecting mathematical reality. The entire argument needs to come out of the theory and its consistancy. Now though the choice of ZFC compatible with large cardinals seems unmotivated, and in fact the use of ZFC seems questionable as well.

Presumably the same argument should work with ZFC+~large cardinals or ZFC+V=L. Yet for these theories the V_i will not provide models. If your response is that the large cardinals still prove they have other models then it would seem you should be equally happy with other conditions which naturally imply the existance of these models, i.e., they don’t need to be V_is. But in this case being well founded gets you everything this argument can give and no support for LCs is added. In other words your argument is subtly circular.

Moreover, I don’t really see the point of all this. If you aren’t a platonist all you need to know is LCs are consistant and produce interesting mathematics. To ‘argue’ for them in this fashion, i.e., doing anything but presenting results and saying isn’t this good math, seems to deeply misunderstand the semantics in question.

9 12 2005

Actually, the elementary equivalence property you mention is much weaker–it follows from the existence of an inaccessible k that there is a club of ordinals so that each Va≤Vk. (It’s an exercise in Jech, actually–the proof is basically that you start anywhere below k and take a union of a sequence of Skolem hulls.)

10 12 2005

Henry – that’s an intriguing point. I’ve found that exercise 12.12 (p. 171) in Jech. I see why the set of such a is closed (because the union of an elementary chain is also elementarily equivalent to each), and I think I understand the Skolem function stuff well enough to realize how to find such an a now. The point is that we need the inaccessible in order to give us a model in which to do all this, since we can define satisfaction relative to that set-model, but not truth in the universe. It still seems surprising to me that a can have countable cofinality and still be such that Va is a model of ZFC. I suppose any such a has to be a strong limit cardinal, because if it wasn’t, then there would be something with rank less than a whose powerset would have actual cardinality greater than a, and this submodel could only assign it a larger cardinality, not a smaller one. But I suppose Va doesn’t realize that a is a singular cardinal, because no cofinal sequence has rank less than a. Anyway, the existence of all these Va models explains nicely why Vk satisfies each chain of consistency statements (apparently including chains of arbitrary length less than k)! Now I’m going to have to read most of Chapter 12 of Jech to figure out the rest of the associated technicalities again.

10 12 2005

Peter – I’ll deal with your last point first. You seem to be presupposing that if one is not a platonist, then one’s only judgement on a set of axioms can be about what logical consequences it has (ie, whether it’s consistent, whether it proves certain “interesting” theorems, etc). However, it seems to me that the natural way to read Field’s position in Science Without Numbers is as providing a uniform, fictional semantics for mathematical statements. That is, there is one preferred “fiction of mathematics”, and we can judge statements as being true in the fiction, false in the fiction, or possibly neither. Thus, in addition to asking whether a set of axioms is consistent or not, and whether it proves a certain result or not, we can ask whether each of its statements is true in the fiction. The axioms of ZFC are postulated to be true in the fiction, but beyond that, we have a variety of means of coming to know what else is true in the fiction. One such means is logical deduction. It seems that (perhaps suitably limited) inference to the best explanation may be another such means, just as we take it to be a means of coming to know what else is true in the real world. Of course, it’s a defeasible method of inference, unlike logical deduction. I wouldn’t let students in 12a reason from “B” and “A->B” to “A”, because I’m trying to teach them about logical deduction. However, I’m happy to let scientists reason from “Mercury’s perihelion shift is x”, and “Einstein’s theory predicts Mercury’s perihelion shift to be x” to “Einstein’s theory is true (or at least a good one to adopt)”, in the absence of countervailing arguments. Using methods like these, we might be able to come to the conclusion that some theory T that is stronger than ZFC is not just consistent and interesting, but also true in the fiction of mathematics.

Similarly, an intuitionist or constructivist might deny the independent existence of mathematical entities, but accept that she can in fact always construct enough things to make any instance of any axiom of ZFC true. I’m not sure if my sort of argument would hold much weight for her, but it does seem that she could well believe in the possibility of more constructions than just those that account for ZFC, and that this would give her reason to adopt further axioms, possibly including certain large cardinals.

You don’t need to be a platonist to move beyond ZFC. If the alternative is just pure formalism, then maybe you’re right that our only judgements about sets of axioms will regard what they prove. But some people think this is right even for platonists (see eg. Ed Zalta and Bernard Linsky’s “Naturalized Platonism vs. Platonized Naturalism”, and Mark Balaguer’s Platonism and Anti-Platonism in Mathematics).

I’m not going to deal here with the question of whether ZFC is well-motivated – I’m just trying to show that if one adopts ZFC, then on a variety of interpretations of mathematics (including certain platonist and fictionalist ones), one should adopt certain large cardinal axioms as well.

Anyway, my premises say nothing in themselves about well-founded models – they just say that the Gödel coding relation properly codes all and only the statements of our language. For this to be true of the Gödel coding relation relative to a model would be for that model to be well-founded. But I’m not talking about coding relative to a model, but the absolute coding relation. The universe (or fictional universe, or whatever) is not a model.

Now, if we believe some set of statements is consistent, then we think our universe should contain a model for it. In particular, the set of statements in ZFC, together with all the facts true about the natural numbers, should be consistent, and therefore there should be a model for them.

Your point about the negation of a large cardinal, or V=L, seems to be misplaced. All of these theories are consistent. If we’ve got an inaccessible, then calling the smallest one I, we see that VI is a model of ZFC+no large cardinals. And LI is a model of V=L that has the proper set of natural numbers. We get all this for free when we get large cardinals. We don’t even need independent motivation to believe these things are consistent – we find natural models right away, once we’ve got natural enough models of ZFC.

Why do we need platonism to say that VI accurately reflects mathematical reality? We accord the same status to this set as we do to any other set, and its got all the same natural numbers and small ordinals as our universe. All arithmetic statements are preserved there. It’s true that there might be non-well-founded models of all these theories as well, but it somehow would seem coincidental if these models just happened to have sets of “natural numbers” that are elementary-equivalent to the actual set. If they’ve got the actual set, then things make sense. So (other things being equal) they should be well-founded, to have the actual set of natural numbers.

10 12 2005

Actually, I see that we don’t need to define satisfaction relative to Vk for an inaccessible k. All we need to do is show that for each formula p, there is a closed unbounded subset of k such that Va reflects p from Vk. This gives us a countable collection of closed unbounded sets, so their intersection is also closed unbounded, and this is a set of a such that Va reflects everything from Vk, and is thus elementary equivalent to it.

We can’t do this for the universe, because we would get a countable collection of closed unbounded classes, and we don’t have any uniform way of talking about an infinite collection of classes and showing that its intersection will also be non-empty.

So now I see that just because Va models ZFC doesn’t imply that a is inaccessible (because in Vk, there are many such a, none of which are inaccessible). So I guess that’s an even weaker assumption than the existence of an inaccessible.

14 12 2005

Woa…I’m even less convinced by your justification now that you explain it better.

Perhaps I’m misreading your statements about fictionalism but as I see it you are just saying that we have some intuitive idea of what sets are trying to look like (say the cumulative hierarchy model) and the question is what axioms best fit this model. That seems fine and reasonable but now I don’t see what this has to do with your argument at all. I suspect you might be able to support LC axioms based on some widespread intuition that sets should go “all the way up”. However, I am at a loss as to how your arguments about reflection help here (see below for the standard argument you just rejected above)

First two key points:

1) V is well founded is really just another way to say that w in V is the real integers (because V believes itself to be well founded). If I somehow turn out to be wrong about my math then eliminate every statement about being well founded and replace it with getting the integers right.

2) Unless you have come up with some fancy new coding relation coding represents only and all proofs iff V gets w right. A proof is a ‘finite’ sequence of statements satisfying certain properties. If we have a non-standard w this means we have non-finite sequences which count as proofs.

3) If V gets w right then all your Con statements are automatically true.

Now no matter what your philosophy of math happens to be (unless you are a full blown formalist who doesn’t believe in *any* second order properties) part of your picture of sets is that the set w *really is* the natural numbers. So take any true number theoretic statement translate it into a statement about w in set theory and as long as you aren’t a full on formalist you should be prepared to accept this as an axiom of set theory. In other words if any of your arguments have any merit we should have already added Con(ZFC), Con(Con(ZFC)) etc.. as axioms and your reasons for large cardinals disappear.

In fact it seems the entire force of your argument disappears if we don’t assume actual set theory is well founded. If actual set theory is not well founded then the set w is not the real natural numbers and hence the satisfaction relation (and proof relation) do not code actual satisfaction (or proof). In other words the statements Con(ZFC) etc.. just become unimportant combinatorial properties and it is kinda misleading to call them Con(ZFC) and suggest they really represent consistancy and the same happens with the statement V_k models ZFC. If we aren’t assuming V is actually well founded modelling is just some weird combinatorial property and why should we take it to be anything special.

Note nothing in the above supposes platonism. When I say ‘actually well founded’ this doesn’t require that their really be objects and some objective notion of being well-founded. It just means in our fiction we take the sets to get w right. When we reason about fictional objects we do something like suppose they are real and see what would be true about them, and I take it this is what you are trying to do, so I don’t see how you can avoid this argument by insisting you are somewhere between platonism and formalism.

Basically your argument goes like this:
1)Assume ZFC is consistant
2)Assume sets can code proofs and satisfaction correctly
3)Therefore sets should model Con(ZFC), Con(Con(ZFC)) etc.. because they are true.

Therefore we have good reason to believe in LC axioms because they make sure set theory gets things right. The problem with this is that assumption 2 already has a nice mathematical characterization we independently think should be true of our sets (V gets the integers right) so we have no need of LC.

14 12 2005

(Ohh I just noticed I think you use actual well foundedness when you make your arguments about large cardinals make Con(Con )of stuff true)

Continuing my argument:

Alright this brings us to the “why use ZFC?” question I raised. Basically I’m asking why we should want models of ZFC in the V_ks. It surely can’t be just to prove the countable collection of consistancy statements you mention even in a compact form. If this was the goal at best your argument would establish that we should add Con(LC) to our language as this number theoretic statement entails all the Con results you mentioned. Likely even we can find “nice” number theoretic statements of equivalent or stronger force (variations of Paris-Harrington theorem or whatnot).

So what is the motivation to have the V_ks model these various con statements? Why not just be happy with the models created by the number theoretic axioms I mentioned above? The standard response from large cardinal advocates is to cite some intuition about initial segments reflecting arbitrarily much mathematical truth (or just the pure elegance of the system which I will address below). The point of my why ZFC why not ZFC+~LC was to defeat this move. If the goal of LCs is to capture what is true of sets in their models then the argument only goes through if we already assume we should not be adopting an axiom incompatible with LCs.

Finally I just don’t see how your argument establishes what you want. Perhaps I was a little dismissive talking about 12a and implication but the point still stands. You have merely shown that LCs imply something you want to be true. Yes in some cases implying a true thing can be evidence in favor of something but not in general. So you need to provide a principle which authorizes the inference in this case!

If your principle is just that mathematicians want elegant powerful axioms and this is just an example of their power then I’m mostly okay with that. Though I think the above considerations make this particular argument less than compelling the consequences of LC for projective determinacy would make the grade (though I still think the basic intuition that drove LC axioms in the first place would be the strongest argument). However, this is not longer really an argument for the axioms any more than a movie review is an argument for you to like the movie…both merely note features that frequently make people find them asthetically appealing but the ultimate judge of the issue is empirical fact not philosophical argument.

I will try and put this all together in a more coherent form on my blog.

14 12 2005

Ohh and on my last point from before and your first point in response. Admittedly I was hasty in what I said (it was clearly wrong) but I think the idea behind it is still correct.

Except insofar as justificaiton of the initial story goes fictionalism is equivalent to platonism for justifying new axioms. The essence of a fictional account is to treat things as if they were true (more or less the way we would a counterfactual though the details get tough here) and then see what else is true.

Since I have a good argument for your argument not going through if we are platonists then this same argument goes through when we fictionally suppose there are these set things.

15 12 2005

OK, that last point is a large part of what I’m trying to say – there’s not much reason why the fictionalist and platonist should have any difference in the set theory they believe, once they’ve both gotten started. At least, to the extent the arguments they use to continue are this sort rather than actual physical applicability issues.

I certainly want V to be well-founded – I was just saying that V isn’t a model – it’s the entire universe (or fictional universe, or whatever). You’re right that having the correct naturals automatically gets me Con(T) for any recursively axiomatizable T that is actually consistent – but I can’t just say “V has the correct naturals” and use this as an axiom, because that statement isn’t formalizable. I can either add all the Con statements directly, or add something like “there is an inaccessible” to get large numbers of them indirectly.

The fact that the complete theory of N isn’t recursively axiomatizable shows that no set of statements of the form Con(T) exhausts the information in N. If ZFC and the complete theory of N are both true (or fictional), then this non-recursively-axiomatizable combination is itself consistent. This statement itself can’t be formalized, but we can say that there is such a model – the easiest way is to say that there is an a such that V_a satisfies ZFC. Once we have that, we have reason to add this to ZFC and continue, saying there are two such cardinals, and then we continue up the chain. But as Henry pointed out, even this is in fact weaker than an inaccessible. The inaccessible gets you consistency statements about further non-recursively-axiomatizable theories that are intuitively consistent.

It looks like I’ll have to be a bit more careful to actually get an inaccessible, but whatever bootstrapping on the V_a’s that model ZFC gets me to an inaccessible, will most likely get me from inaccessibles to Mahlo’s, and then to hyper-Mahlo’s, and so on. Though the general form of argument will have to change to get to a measurable.

15 12 2005

Basically, I’m saying that there are non-recursively-axiomatizable theories that are consistent. The only way to guarantee our universe “realizes” this is to make sure that there are models. I suppose the weakest way to say this is just to say “there is a well-founded model of ZFC” (which is of course stronger than Con(ZFC)), and then we start going up the hierarchy of those statements (saying such models are nested). With just a few facts about the powerset relation thrown in, we may have to say “some V_a is a model of ZFC”, and we go up that hierarchy. Some statement about these models having the full strength of replacement might be needed to get to an inaccessible, but we’re certainly on the way.

16 12 2005

First of all I don’t see where you show that non-recursively axiomatizable theories are consistant. The collection Con(ZFC), Con(Con(ZFC)) etc.. is recursively axiomitizeable. Since every r.e. set of axioms is actually recursively axiomatizeable if you can enumerate these axioms they are actually recursively axiomitzeable. By Church’s thesis if you can intuitively explain how we can list off these theories (take the last one and say it is consistant) it is recursively axiomitizeable.

If on the other hand what you want is to prove that Th(N)+ZFC is itself consistant then you are correct that no finite (or even recursively axiomatizeable) list of number theoretic (really set theoretic statements only talking about w) can accomplish this. However, neither can any recursively axiomatizeable extension of ZFC and since you can’t even really describe a non-recusive axiomitization I presume you aren’t advocating we adopt some non-recursive list of LC axioms.

The easy proof of this is as follows. If we can prove that ZFC+Th(N) has a model we can prove that Th(N) has a model (just chop off the first part of our model and take restrictions of the appropriate formulas). So we can focus on showing no none r.e. extension of ZFC can show the existance of a model of Th(N). Now let the theory complete with LC axioms or whatever be denoted T, if T is inconsistant we can throw it out right away and if not Con(T) is an element of Th(N). However, since T is recursively axiomitized the statement Con(T) is pi-1 and hence if true in model M it is also true in any submodel N of M and in fact this is provable in ZFC. Since it is also true in ZFC that N is a submodel of anything modeling PA it follows that proving you have a model of Th(N) entails you have proved Con(T) is true in the actual w. Hence by Godel you can’t prove the existance of any model of Th(N).

So things look exactly the same if we strengthen things by adding new number theoretic truths or if we add new LC axioms. In either case we can add new axioms to guarantee the consistancy of larger and larger classes of theories but in either case we can never finish with any finite or recursive set of axioms.

More generally the point you miss is that we can perfectly well talk about non-recursive theories restricting our attention to w. If we can talk about the theory in your big set theory that means we can find a formula phi(x) s.t. phi(x) intuitively holds iff x is a code of an axiom of the theory in question. The statement of consistancy for the theory represented by phi(x) is just that ~Ey:Ax coded in y x follows from earlier codes in y OR phi(x). These are all statements which can be simply added to the theory directly rather than bringing LCs into the picture.

17 12 2005

A minor note on one of Peter’s comments above (12/14 5:46). At a couple of places, you seem to suggest that well-foundedness is the same as getting w right. In general, this isn’t true. It’s easy enough to build non-well-founded models of ZFC which still get w right (the ordinals just go haywire higher up).

Of course, in context, this may not be an issue, as getting w right is the only part of well-foundedness that’s relevant to discussing things like consistency statements. Still, it’s worth noting.