# Re: Paper and slides on indefiniteness of CH

This is a continuation of my earlier message. Recall that I have two titles to this note. You get to pick the title that you want.

REFUTATION OF THE CONTINUUM HYPOTHESIS AND EXTENDIBLE CARDINALS

THE PITFALLS OF CITING “INTRINSIC MAXIMALITY”

1. GENERAL STRATEGY.
2. THE LANGUAGE $L_0$.
3. STRONGER LANGUAGES.

1. GENERAL STRATEGY

Here we present a way of using the informal idea of “intrinsic maximality of the set theoretic universe” to do two things:

1. Refute the continuum hypothesis (using PD and less).
2. Refute the existence of extendible cardinals (in ZFC).

Quite a tall order!

Since I am not that comfortable with “intrinsic maximality”, I am happy to view this for the time being as an additional reason to be even less comfortable.

At least I will resist announcing that I have refuted both the continuum hypothesis and existence of certain extensitvely studied large cardinals!

INFORMAL HYPOTHESIS. Let $\phi(x,y,z)$ be a simple property of sets $x,y,z$. Suppose ZFC + “for all infinite $x$, there exist infinitely many distinct sets which are pairwise incomparable under $\phi(x,y,z)$” is consistent. Then for all infinite $x$, there exist infinitely many distinct sets which are pairwise incomparable under $\phi(x,y,z)$.

Since we are going to be considering only very simple properties, we allow for more flexibility.

INFORMAL HYPOTHESIS. Let $0 \leq n,m \leq \omega$. Let $\phi(x,y,z)$ be a simple property of sets $x,y,z$. Suppose ZFC + “for all $x$ with at least $n$ elements, there exist $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$” is consistent. Then for all $x$ with at least $n$ elements, there exist at least $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$.

We can view the above as reflecting the “intrinsic maximality of the set theoretic universe”.

We will see that this Informal Hypothesis leads to “refutations” of both the continuum hypothesis and the existence of certain large cardinals, even using very primitive phi in very primitive set theoretic languages.

2. THE LANGUAGE $L_0$

$L_0$ has variables over sets, $=$,$<$, $\leq^*$,$\cup$. Here $=$,$<$, $=^*$ are binary relation symbols, and $\cup$ is a unary function symbol. $x \leq^* y$ is interpreted as “there exists a function from $x$ onto $y$“. $\cup$ is the usual union operator, $\cup x$ being the set of all elements of elements of x.

$\text{MAX}(L_0,n,m)$. Let $0 \leq n,m \leq \omega$. Let $\phi(x,y,z)$ be the conjunction of finitely many formulas of $L_0$ in variables $x,y,z$. Suppose ZFC + “for all $x$ with at least $n$ elements, there exist $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$” is consistent. Then for all $x$ with at least $n$ elements, there exist at least $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$.

THEOREM 2.1. ZFC + $\text{MAX}(L_0,\omega,\omega)$ proves that there is no $(\omega+2)$-extendible cardinal.

More generally, we have

THEOREM 2.2. Let $2 < \log(m)+1 < n \leq \omega$.

i. ZFC + $\text{MAX}(L_0,n,m)$ proves that there is no $(\omega+2)$-extendible cardinal. Here $\log(\omega) = \omega$.
ii. ZFC + PD + $\text{MAX}(L_0,n,m)$ proves that the GCH fails at all infinite cardinals. In particular, it refutes the continuum hypothesis.
iii. ii with PD replaced by higher order measurable cardinals in the sense of Mitchell.

We are morally certain that we can easily get a complete understanding of the meaning of the sentences in quotes that arise in the $\text{MAX}(L_0,n,m)$.

Write $\text{MAX}(L_0)$ for

“For all $0 \leq n,m \leq \omega$, $\text{MAX}(L_0,n,m)$“. Using such a complete understanding we should be able to establish that ZFC + $\text{MAX}(L_0)$ is a “good theory”. E.g., such things as

1. ZFC + PD + $\text{MAX}(L_0)$ is equiconsistent with ZFC + PD.
2. ZFC + PD + $\text{MAX}(L_0)$ is conservative over ZFC + PD for sentences of second order arithmetic.
3. ZFC + PD + $\text{MAX}(L_0)$ + “there is a proper class of measurable cardinals” is also conservative over ZFC + PD for sentences of second order arithmetic.

We will revisit this development after we have gained that complete understanding. Then we will go beyond finite conjunctions of atomic formulas in $L_0$.

The key technical ingredient in this development is the fact that

1. GCH fails at all infinite cardinals is incompatible with $(\omega+2)$-extendible cardinals (Solovay).
2. GCH fails at all infinite cardinals is demonstrably consistent using much weaker large cardinals, or using just PD (Foreman/Woodin).

Harvey

# Re: Paper and slides on indefiniteness of CH

I have two titles to this note. You get to pick the title that you want.

REFUTATION OF THE CONTINUUM HYPOTHESIS THE PITFALLS OF CITING “INTRINSIC MAXIMALITY’

Note that the most fundamental and simple nontrivial equivalence relation on the set theoretic universe is that of “being in one-one correspondence”.

Also very fundamental and simple is the equivalence relation EQ on infinite sets of reals “being in one-to-one correspondence”.

Note that it is consistent with ZFC that this fundamental simple EQ has

i. exactly two equivalence classes. ii. infinitely many equivalence classes.

THEREFORE, by the “intrinsic maximality of the set theoretic universe”, ii holds. THEREFORE, we have refuted the continuum hypothesis (smile).

NOTE: A lesson that can be drawn here is just how important it is to avoid cavalier quoting of “intrinsic maximality of the set theoretic universe”.

In fact, if we factor, we are looking at a set for which it is consistent with ZFC that it has, on the one hand, exactly two elements, and on the other hand, is infinite. So by “intrinsic maximality of the set theoretic universe”, it must be infinite (smile).

GENERAL PRINCIPLE. Let EQ be a simple equivalence relation. Suppose ZFC + “EQ has infinitely many equivalence classes” is consistent. Then EQ actually has infinitely many equivalence classes.

Here is the legitimate foundational program.

1. Set up an elementary language that is based on only some of the most set theoretically fundamental notions.
2. Determine which “simple” definitions define equivalence relations. Show that this is robust, in that here truth is the same as provability in ZFC and in ZC.
3. Determine what is consistent with ZFC about the number of equivalence classes of items in 2.
4. Now apply the general principle, and show that the resulting statements are (even collectively?) consistent with ZFC. Perhaps the general principle will be seen to be equivalent over ZFC to “the continuum is greater than $\aleph_\omega$” or perhaps some versions of not GCH?
5. Rework 1-4 with ever stronger elementary languages and ever less “simple” definitions, until one hits a brick wall.

The immediate problem is to get a good prototype for this elementary language. We want it to be not ad hoc, and so should be in tune with the most basic set theoretic material.

While doing this real time foundations, it now appears, provisionally, that we are best off using “there is a function from x onto y” and not just “there is a bijection from x onto y”. The former is more flexible than the latter, and still very very basic for elementary set theory.

ELST = elementary set theory. We have

1. Equality, and union operator (set of all elements of elements).
2. There is a function from $x$ onto $y$. Written $x\geq y$.
3. Convenient to have variables range only over infinite sets.

Something interesting has arisen. This language supports even more naturally the 3-ary relation

$T(x,y,z)$ if and only if

i. The union of y and the union of z are both x.
ii. $y \geq z\geq x$ and $z \geq y \geq x$.
iii. (Implicitly, x,y,z are infinite).

Two observations.

1. We have defined T as a conjunction of a small number of atomic formulas in $x,y,z,$ with no nesting of the union operator.
2. For all $x$, $T_x$ is an equivalence relation.

Thus there is great simplicity here. We can provisionally concentrate on just cases of 1.2, even perhaps with a limit on the number of atomic formulas. We can also relax the “no nesting”.

So we have a parameterized equivalence relation. We should look at a modified General Principle.

GENERAL PRINCIPLE. Let T be a 3-ary parameterized equivalence relation. Suppose ZFC + “EQ has infinitely many equivalence classes” is consistent. Then EQ actually has infinitely many equivalence classes.

In this way, we should be getting the robustness referred to above, and also the failure of GCH at every infinite cardinal.

I’ll stop here with this provisional beginning…

Harvey

# Re: Paper and slides on indefiniteness of CH

Mr. Energy writes (two excerpts):

With co-authors I established the consistency of the following

Maximality Criterion. For each infinite cardinal $\alpha$, $\alpha^+$ of $\text{HOD}$ is less than $\alpha^+$.

Both Hugh and I feel that this Criterion violates the existence of certain large cardinals. If that is confirmed, then I will (tentatively) conclude that Maximality contradicts the existence of large cardinals. Hugh will conclude that there is something wrong with the above Maximality Criterion and it therefore should be rejected.

My point is that Hugh considers large cardinal existence to be part of set-theoretic truth. Why? I have yet to see an argument that large cardinal existence is needed for “good set theory”, so it does not follow from Type 1 evidence. That is why I think that large cardinal existence is part of Hugh’s personal theory of truth.

My guess is he’d also consider type 2 evidence (involving the relations of set theory to the rest of mathematics) if there were some ready to hand.

There is some ready to hand: At present, Type 2 evidence points towards Forcing Axioms, and these contradict CH and therefore contradict Ultimate L

I have written dozens of e-mails to explain what I am doing and I take it as a good sign that I am still standing, having responded consistently to each point. If there is something genuinely new to be said, fine, I will respond to it, but as I see it now we have covered everything: The HP is simply a focused investigation of mathematical criteria for the maximality of V in height and width, with the aim of convergence towards an optimal such criterion. The success of the programme will be judged by the extent to which it achieves that goal. Interesting math has already come out of the programme and will continue to come out of it. I am glad that at least Hugh has offered a bit of encouragement to me to get to work on it.

This illustrates the pitfalls involved in trying to use an idiosyncratic propogandistic slogan like “HP” to refer to an unanalyzed philosophical conception with language like “intrinsic maximality of the set theoretic universe”. Just look at how treacherous this whole area of “philosophically motivated higher set theory” can be.

E.g., MA (Martin’s axiom) already under appropriate formulations look like some sort of “intrinsic maximality”, at least as clear as many things purported on this thread to exhibit some sort of “intrinsic maximality”, and already implies that CH is false. So have we now completely solved the CH negatively? If so, why? If not, why not? See what happens with an unanalyzed notion of “intrinsic maximality of the set theoretic universe”. Also MM (Martin’s maximum) is even stronger, and implies that $2^\omega = \omega_2$. Also looks like “intrinsic maximality of the set theoretic universe”, at least before any convincing analysis of it, and so do we now know that $2^\omega = \omega_2$ follows from the “intrinsic maximality of the set theoretic universe”?

I will now take an obvious step toward turning at least some of this very unsatisfying stuff into something completely unproblematic – without the idiosyncratic propogandistic slogans – AND something (hopefully) not needing countable transitive models for straightforward formulations.

1. We want to explore the idea that

*L is a tiny part of V* *L is very different from V*

We also want to explore the idea that

**HOD is a tiny part of V. **HOD is very different from V**

Here HOD = hereditarily ordinal definable sets. Myhill/Scott proved that HOD satisfies ZFC, following semiformal remarks of Gödel.

2. There are some interesting arguments that one can give for L being a tiny part of V. These arguments themselves can be subjected to various kinds of scrutiny, and that is an interesting topic in and of its own. But we shall, for the time being, take it for granted that we are starting off with “L is a tiny part of V”.

3. On the other hand, the arguments that HOD is a tiny part of V are, at least at the moment, fewer and much weaker. This reflects some important technical differences between L and HOD. E.g., L is very stable in the sense that L within L is L. However, HOD within HOD may not be HOD (that’s independent of ZFC).

4. Another related big difference between L and HOD is the following. You can prove that any formal extension of the set theoretic universe compatible with the set theoretic universe in a nice sense, must violate V = L if the original set theoretic universe violates V = L. This is the kind of thing that adds to an arsenal of possible arguments that L is only a part or tiny part of V. However, the set theoretic universe demonstrably has a formal extension satisfying V = HOD even if the set theoretic universe does not satisfy V = HOD. This makes the idea that HOD is a tiny part of V a much more problematic “consequence” of “intrinsic maximality of the set theoretic universe”.

5. Yet another difference. Vopenka proved in ZFC that every set can be obtained by set forcing over HOD. That every set can be obtained by set forcing over L is known to be independent of ZFC, and in fact violates medium large cardinals (such as measurable cardinals and even $0^\#$). The same is true for set forcing replaced by class forcing.

6. Incidentally, I think there is an open question that goes something like this. Let M be the minimum ctm of ZFC. There exists a ctm extension of M with the same ordinals that is not obtainable by class forcing over M – I think even under a very wide notion of class forcing. Still open?

7. Another way of talking about the problematic nature of V not equal HOD as following from “intrinsic maximality” is that, well, maybe if there were more sets, we would be able to make more powerful definitions, making certain certain sets in HOD that weren’t “before”, and then close this off, making V = HOD. Thus this is an attempt to actually turn V = HOD itself into some sort of “intrinsic maximality”!!

8. So the proper move, until there is more creative analysis of “intrinsic maximality of the set theoretic universe” is to simply say, flat out:

*we are going to explore the idea that HOD is a tiny part of V* *we are going to explore the idea that HOD is very different from V*

and avoid any idiosnyncratic propogandistic slogans like “HP”.

9. So now let’s fast forward to the excerpt from Mr. Energy:

With co-authors I established the consistency of the following Maximality Criterion. For each infinite cardinal $\alpha$, $\alpha^+$ of HOD is less than $\alpha^+$. Both Hugh and I feel that this Criterion violates the existence of certain large cardinals. If that is confirmed, then I will (tentatively) conclude that Maximality contradicts the existence of large cardinals. Hugh will conclude that there is something wrong with the above Maximality Criterion and it therefore should be rejected.

Here is a reasonable restatement without the idiosyncratic propoganda – propoganda that papers over all of the issues about HOD raised above.

NEW STATEMENT. With co-authors I (Mr. Energy) established the consistency of the following relative to the consistency of ???

(HOD very different from V). Every infinite set in HOD is the domain of a bijection onto another set in HOD without there being a bijection in HOD.

Furthermore, Hugh and I (Mr. Energy) feel that the above statement refutes the existence of certain kinds of large cardinal hypotheses. If this is confirmed, then it follows that “HOD is very different from V” is incompatible with certain kinds of large cardinal hypotheses.

10. Who can complain about that? Perhaps somebody on the list can clarify just which large cardinal hypotheses might be incompatible with the above statement?

11. Let’s now step back and reflect on this a bit in general terms to make more of it. What can be say about “HOD very different from V” in general terms?

HOD is an elementary substructure of V

is of course very strong. This is equivalent to saying that V = HOD.

But the above statement is an extremely strong refutation of elementary substructurehood.

THEOREM (?). The most severe/simplest possible violation of L being an elementary substructure of V is that “every infinite set in L is the domain of a bijection onto another set in L without there being a bijection in L”.

THEOREM (?). The most severe/simplest possible violation of HOD being an elementary substructure of V is that “every infinite set in HOD is the domain of a bijection onto another set in HOD without there being a bijection in HOD”.

THEOREM (???). The most severe/simplest possible violation of V not equaled to L is that “every infinite set in L is the domain of a bijection onto another set in L without there being a bijection in L”.

THEOREM (???). The most severe/simplest possible violation of V not equaled to HOD is that “every infinite set in HOD is the domain of a bijection onto another set in HOD without there being a bijection in HOD”.

Since this morning I am doing some real time foundations (of higher set theory), I should be allowed to state Theorems without knowing how to state them.

I also reserve the right to stop here.

I have written dozens of e-mails to explain what I am doing and I take it as a good sign that I am still standing, having responded consistently to each point. If there is something genuinely new to be said, fine, I will respond to it, but as I see it now we have covered everything: The HP is simply a focused investigation of mathematical criteria for the maximality of V in height and width, with the aim of convergence towards an optimal such criterion. The success of the programme will be judged by the extent to which it achieves that goal. Interesting math has already come out of the programme and will continue to come out of it. I am glad that at least Hugh has offered a bit of encouragement to me to get to work on it.

Of course, you have chosen to respond to much but not all of what everybody has written here, except me, invoking the “brother privilege”. Actually, I wonder if the “brother privilege” – that you do not have to respond to your brother in an open intellectual forum – is a consequence of the “intrinsic maximality of the set theoretic universe”?

If you are looking for “something genuinely new to say” then you can start with the dozens of emails I have put on this thread, Actually, you have covered very little by serious foundational standards.

On a mathematical note, you can start by talking about #-generation, what it means in generally understandable terms, why it is natural and/or important, and so forth. Why it is an appropriate vehicle for “fixing” IMH (if it is). It is absurd to think that a two line description weeks (or is it months) ago is even remotely appropriate for a list of about 75 readers. Also, continually referring to type 1, type 2, type 3 set theoretic themes without using real and short names is a totally unnecessary abuse of the readers of this list. People are generally not going to be keeping that in their heads – even if they have not been throwing your messages (and mine) into the trash. Are the numbers 1,2,3 canonically associated with those themes? Furthermore, your brief discussion of them was entirely superficial. There are crucial issues involved in just what the interaction of higher set theory is with mathematics that have not been discussed hardly at all here either by you or by others.

Change HP to CTMP = countable transitive model program. Cast headlines for statements in terms like “HOD is very different from V” or “HOD is a tiny part of V” or things like that. Avoid “intrinsic maximality of the set theoretic universe” unless you have something new to say that is philosophically compelling.

Harvey

# Re: Paper and slides on indefiniteness of CH

Dear Peter and Harvey,

On Sun, 26 Oct 2014, Koellner, Peter wrote:

Dear Harvey …

Is $\textsf{IMH}^\#$ merely a layering of the $\textsf{IMH}$ idea on top of large cardinal infrastructure?

Sy gave a precise definition of this — see e.g. his letter to Bob for the definition of #-generation.

Thank you, Peter, for pointing this out, but what I said to Bob on 25.September was only the definition of #-generation. Below I repeat the motivation for the $\textsf{IMH}^\#$ (called IMH(#-generation) below), which I communicated to Pen on 7.August. In reference to my BSL paper with Tatiana entitled “The Hyperuniverse Programme” I said:

What has changed in my perspective since the BSL paper (I cannot speak for Tatiana) regards the “ultimate” nature of what the programme reveals about truth and the relationship between the programme and set-theoretic practice. Penny, you are perfectly right to ask:

Is it really essential that these statements be ‘ultimate and unrevisable’?  Isn’t it enough that they’re the ones we accept for now, reserving the right to adjust our thinking as we learn more?

At the time we wrote the paper we were thinking almost exclusively of the IMH, which contradicts the existence of inaccessible cardinals. This is of course a shocking outcome of a reasoned procedure based on the concept of “maximality”! This caused us to rethink the role of large cardinals in set-theoretic practice and to support the conclusion that in fact the importance of large cardinals in set theoretic practice derives from their existence in inner models, not in V. Indeed, I still support that conclusion and on that basis Tatiana and I were prepared to declare the first-order consequences of the IMH as being ultimate truths.

But what I came to realise is that the IMH deals only with “powerset maximality” and it is compelling to also introduce “ordinal maximality” into the picture. (I should have come to that conclusion earlier, as indeed the existence of inaccessible cardinals is derivable from the intrinsic maximal iterative concept of set!) There are various ways to formalise ordinal maximality as a mathematical criterion … [a strong] form due to Honzik and myself is #-generation, which roughly speaking asserts the existence of any large cardinal notion compatible with V = L…we can consistently formulate the synthesis IMH(#-generation) of IMH with #-generation…the surprise is that IMH(#-generation) is a synthesised form of powerset maximality with ordinal maximality which is compatible with all large cardinals (even supercompacts!), and one can argue that #-generation is the “correct” mathematical formulation of ordinal maximality.

This was an important lesson for me and strongly confirms what you suggested: In the HP (Hyperuniverse Programme) we are not able to declare ultimate and unrevisable truths. Instead it is a dynamic process of exploration of the different ways of instantiating intrinsic features of universes. learning their consequences and synthesising criteria together with the long-term goal of converging towards a stable notion of “preferred universe”.

NOTE: I had concluded that, on the basis of this extensive traffic, this “HP” is not a legitimate foundational program, and should be renamed CTMP = countable transitive model program,

Once again, as I wrote (in various formulations) not only to Peter (26.October), but also to Pen and Geoffrey (23.October) and to Neil (25.October):

Now I can be even more accomodating. Some of you doubters out there may buy the way I propose to treat maximality via a Single-Universe view (via lengthenings and “thickenings”) but hide your money when it comes to the “reduction to the Hyperuniverse” … OK, then i would say the following … Fine, forget about the reduction to countable transitive models, just stay with the (awkward) way of analysing maximality that I describe above (via lengthenings and “thickenings”) without leaving “the real V”! You don’t need to move the discussion to countable transitive models anyway, it was just what I considered to be a convenience of great clarification-power, nothing more!

Is everybody happy now? You can have your “real V” and you don’t need to talk about countable transitive models of ZFC. What remains is nevertheless a powerful way to discuss and extract consequences from the maximality of V in height and width … [admittedly] you strip the programme of the name “Hyperuniverse Programme” and it becomes the “Maximality Programme” or something like that … [but] it’s only a change of name, not a change of approach or content in the programme.

NOTE: My own position is that “intrinsic maximality of the set theoretic universe” is prima facie a deeply flawed notion, fraught with nonrobustness (inconsistencies, particularly) that may or may not be coherently adjusted in order to lead to anything foundationally interesting.

I have said something similar.

I understand that you, like Pen, hesitate to accept “intrinsic justifications” for new axioms based on the maximal iterative conception (the maximality of V in height and width). Indeed “intrinsic justification” is a very strong phrase.

Pen and I, both in private and in the broader discussion, have reverted to the phrase “intrinsic heuristic”, which is much softer. My question for you is this: Do you regard the fact that a statement follows from maximality “as an intrinsic heuristic” as evidence for its truth (in addition to other forms of evidence for truth coming from the roles of set theory as good mathematics and as a foundation for mathematics)?

Thanks,
Sy

PS: In answer to an earlier question, I am indeed naturally inclined to think in terms of the stronger form of radical potentialism. Indeed I do think that, as with height actualism, there are arguments to suggest that the weaker form of radical potentialism without the stronger form is untenable.