Re: Paper and slides on indefiniteness of CH

Dear Sy,

I write with your permission to summarize for the group a brief exchange we had in private. Before that exchange began, you had agreed to these three points:

  1. The relevant concept is the familiar iterative conception, which includes a rough idea of maximality in ‘height’ and ‘width’.
  2. To give an intrinsic justification or intrinsic evidence for a set-theoretic principle is to show that it is implicit in the concept in (1).
  3. The HP is a method for extracting more of the implicit content of the concept in (1) than has heretofore been possible.

We then set about exploring how the process in (3) is supposed to work, beginning with more careful attention to the iterative conception in (1). You summarize it this way:

“Maximal” means “as large as possible”, whether one is talking about

a. Vertical or ordinal-maximality: the ordinal sequence is “as long as possible”, or about

b. Horizontal or powerset-maximality: the powerset of any set is “as large as possible”.

In other words there is implicitly a “comparative” (and “modal”) aspect to “maximality”, as to be “as large as possible” can only mean “as large as possible within the realm of ‘possibilities'”.

Thus to explain ordinal- and powerset-maximality we need to compare different possible mental pictures of the set-theoretic universe. In the case of ordinal-maximality we need to consider the possibility of two mental pictures P and P* where P* “lengthens” P, i.e. the universe described by P is a rank initial segment of the universe described by P*. We can now begin to explain ordinal-maximality. If a picture P of the universe is ordinal-maximal then any “property” of the universe described by P also holds of a rank initial segment of that universe. This is also called “reflection”.

In the case of powerset maximality we need to consider the possibility of two mental pictures P and P* of the universe where P* “thickens” P, i.e. the universe described by P is a proper inner model of the universe described by P*.

There seemed to me to be something off about a universe being ‘maximal in width’, but also having a ‘thickening’. Citing Peter Koellner’s work, you replied that reflection actually involves ‘lengthenings’ (to which the ‘thickenings’ would be analogous), because it appeals to higher-order logics:

Reflection has the appearance of being “internal” to V, referring only to V and its rank initial segments. But this is a false impression, as “reflection” is normally taken to mean more than 1st-order reflection. Consider 2nd-order reflection (for simplicity without parameters):

({*}) If a 2nd-order sentence holds of V then it holds of some V_\alpha.

This is equivalent to:

({*}{*}) If a 1st-order sentence holds of V_{\text{Ord} + 1} then it holds of some V_{\alpha + 1},

where \text{Ord} denotes the class of ordinals and V_{\text{Ord} + 1} denotes the (3rd-order) collection of classes. In other words, 2nd-order reflection is just 1st-order reflection from V_{\text{Ord} + 1} to some V_{\alpha + 1}. Note that V_{\text{Ord} + 1} is a “lengthening” of V = V_\text{Ord}. Analogously, 3rd order reflection is 1st-order reflection from the lengthening V_{\text{Ord} + 2} to some V_{\alpha + 2}. Stronger forms of reflection refer to longer lengthenings of V.

1st-order forms of reflection do not require lengthenings of V but are very weak, below one inaccessible cardinal. But higher-order forms yield Mahlo cardinals and much more, and this is what Goedel and others had in mind when they spoke of reflection.

Another way of seeing that lengthenings are implicit in reflection is as follows. In its most general form, reflection says:

({*}{*}{*}) If a “property” holds of V then it holds of some V_\alpha.

This is equivalent to:

({*}{*}{*}{*}) If a “property” holds of each V_\alpha then it holds of V.

[({*}{*}{*}) for a "property" is logically equivalent to ({*}{*}{*}{*}) for the negation of that "property".]

OK, now apply ({*}{*}{*}{*}) to the property of having a lengthening that models ZFC. Clearly each V_\alpha has such a lengthening, namely V. So by ({*}{*}{*}{*}), V itself has lengthenings that model ZFC! One can then use this to infer huge amounts of reflection, far past what Goedel was talking about.

I am not assuming that everybody is a “potentialist” about V. Even the Platonist can have mental images of the lengthenings demanded for reflection. And without such lengthenings, reflection has been reduced to a principle weaker than one inaccessible cardinal.

Now given that lengthenings are essential to ordinal-maximality isn’t it clear that thickenings are essential to powerset-maximality? We can then begin to explain powerset-maximality as follows: A picture P of the universe is powerset-maximal if any “property” of the universe described by a thickening of P also holds of the universe described by some thinning of P. What I called the weak-IMH is the “follow your nose” mathematical formulation of this notion of powerset-maximality for first-order properties.

I proposed that we consult with Peter Koellner directly about this business of ‘lengthenings’ in reflection, and this is why we again address the group.

(So, what do you think, Peter?)

Finally, you suggested that you might consider retracting (2) above and returning to the proposal of a different conception of set. The challenge there is to do so without returning to the unappealing idea that ‘intrinsic justification’ and ‘set-theoretic truth’ are determined by a conception of the set-theoretic universe that’s special to a select group.

All best,
Pen

Re: Paper and slides on indefiniteness of CH

I see the importance you are attaching to your Ultimate L Conjecture – particularly getting a proof “by the current scenarios of course”. Care to make rough probabilistic predictions on when you will prove it “by the current scenarios of course”? Until then, your point of view seems to be that statements like Con(HUGE) are wide open, and you currently are not willing to declare any confidence in them.

Fascinating as this is, I think people here might be even more interested in the implications Con(EFA) arrows Con(PA) arrows Con(Z) arrows Con(ZFC) arrows Con(ZFC + measurable) arrows Con(ZFC + PD). Maybe you can comment on at least one of these arrows? — or maybe Peter Koellner?

Based on all my experience to date I have a conception of V in which PD holds. Based on that conception it is impossible for PD to be inconsistent. But that conception may be a false conception.

If it is a false conception then I do not have a conception of V to fall back on except for the naive conception I had when I first was exposed to set theory. This is why for me, if ZFC+PD is inconsistent I think that ZFC is suspect. This is not to say that I cannot or will not rebuild my conception to that of V which satisfies ZFC etc. But I would need to understand how all the intuitions etc. that led me to a conception V with PD went so wrong.

My question to those who feel V is just the integers (and maybe just a bit more) is: How do they assess that Con ZFC+PD is relevant to their conception? The conception of the set theoretic V with or without PD are very different conceptions with deep structural differences. I just do not see that happening yet to anywhere near the same degree in the case where the conception V  is just the integers. But as your work suggests this could well change.

But even so, somehow a structural divergence alone does not seem enough (to declare that Con ZFC+PD is an indispensable part of that conception).  Who knows, maybe there is an arithmetically based strongly motivated hierarchy of “large cardinals” and Con PD matches something there.

If one’s conception of V is the integers and one is never compelled to declare Con ZFC+PD as true based on whatever methodology one is using to refine that conception, then it seems to me that the only plausible conjecture one can make is that ZFC+PD is inconsistent. This was the cryptic point behind item (2) in my message which inspired your press releases.

To those still reading and to those who have participated, I would like to express my appreciation. But I really feel it is time to conclude my participation in this email thread. Classes are about to begin and I shall have to return to Cambridge shortly.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

I see the importance you are attaching to your Ultimate L Conjecture — particularly getting a proof “by the current scenarios of course”. Care to make rough probabilistic predictions on when you will prove it “by the current scenarios of course”? Until then, your point of view seems to be that statements like Con(HUGE) are wide open, and you currently are not willing to declare any confidence in them.
 
Fascinating as this is, I think people here might be even more interested in the implications Con(EFA) arrows Con(PA) arrows Con(Z) arrows Con(ZFC) arrows Con(ZFC + measurable) arrows Con(ZFC + PD). Maybe you can comment on at least one of these arrows? — or maybe Peter Koellner?
 
Harvey

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I think it is a bit presumptuous of me to answer these questions in an email thread with such a large cc. I will however answer (1) since that is quite closely related to issues already discussed.

The short answer to (1) is actually no. Let me explain. If the Ultimate L Conjecture is true than I would certainly move supercompact to the safe zone on equal footing with PD. I do not currently place supercompact there.

What about huge cardinals? Ultimate L will be constructed as the inner model for exactly one supercompact cardinal. But unlike all other inner model constructions, this inner model will be universal for all large cardinal axioms we know of (large cardinals such as huge will be there if they occur in the parent universe within which Ultimate L is constructed but they play no role in the actual construction).

Therefore if the Ultimate L Conjecture is true (and the proof is by the current scenarios of course), inner model theory can no longer provide direct evidence for consistency since the large cardinals past supercompact play no role in the construction of Ultimate L. This is just as for L, the large cardinals compatible with L play no role in the construction of L.

So if the Ultimate L Conjecture is true then there is serious challenge. How does one justify the large cardinals beyond supercompact? My guess is that their justification will involve how they affect the structure of Ultimate L. For example, consider the following conjecture.

Conjecture: Suppose V = Ultimate L. Suppose \lambda is an uncountable cardinal such that the Axiom of Choice fails in L(P(\lambda)). Then there is a non-trivial elementary embedding j:V_{\lambda+1} \to V_{\lambda+1}.

If conjectures such as this are true then it seems very likely that for large cardinals beyond supercompact, their true natures etc., are really only revealed within the setting of V = Ultimate L and it is only in that unveiling that one is able to make the case for consistency. In this scenario, V = Ultimate L is not a limiting axiom at all, it is the axiom by which the true nature of the large cardinal hierarchy is finally uncovered.

But it is important to keep in mind, the Ultimate L Conjecture could be false. Indeed one could reasonably conjecture that it is the conception of a weak extender model which is not correct once one reaches the level of supercompact even though it seems to be correct below that level.

However at present and for me, the Ultimate L Conjecture is the keystone in a very tempting and compelling picture.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

I personally don’t feel the implication

If ZFC + PD is inconsistent then ZFC is inconsistent.

I have a couple of questions for you.

  1. I am under the impression that you are not committed to the consistency of ZFC + HUGE. Are you committed to the consistency of ZFC + LC roughly if and only if there is some good inner model theory for it? Are you also advocating a more general principle of this kind?
  2. Consider the statement: If ZFC is inconsistent then T is inconsistent. For how weak a T do you feel this? As an extreme, are you willing to take T down to EFA = exponential function arithmetic?
  3. As you can see from what I wrote about blurring pictures, my own view is one of relative clarity and therefore relative confidence. But you seem to have quite a different view, and I am wondering what you can say about your view (feelings, intuition)?

Harvey

Re: Paper and slides on indefiniteness of CH

OK here is my press release.

URGENT PRESS RELEASE
FRIED INSTITUTE OF FOM

Professor and founder, H. Fried, stunned the already reeling mathematics community with the improvement of Professor Wood’s celebrated results of last year on the inconsistency of PD. Professor Fried extended Wood’s results to the ZFC axioms themselves.

As widely reported at the time, most experts had hoped that this level would remain safe even light of Wood’s results. Professor Wood however disagreed saying that he had always believed that if PD was inconsistent then one could not be sure that even ZFC itself was safe.  Even so Wood noted, it is remarkable that Fried’s results were obtained so soon after his own results were made public. Wood continued by saying that he had tried very hard to extend his results to ZFC but had been unable to do this. These are exciting times he concludes, stating it is by no means clear where the “safe” area lies.

Honoring a declaration Wood made at the Gödel Centenary meeting, Wood also announced his resignation from Havad.

Re: Paper and slides on indefiniteness of CH

The first Press Release has been amended below according to John Burgess’ suggestion:

URGENT PRESS RELEASE
HAVAD UNIVERSITY

Professor Wood, who recently moved to the Havad mathematics department from Berkeley, has stunned the mathematical and philosophical world with his breathtaking demolition of the standard foundations for mathematics that has been almost universally accepted since the 1920’s. In a development of epoch proportions, Wood has actually shown that the usual ZFC axioms for mathematics are in fact inconsistent. For example, Wood has been able to prove from the ZFC axioms that both 2+2 = 4 and 2+2 = 5.

All experts in the foundations of mathematics interviewed considered this development to be astonishing beyond belief, as it threatens to throw the foundations of mathematics into a complete state of utter chaos. They agreed that the only chance for some calm would be if the inconsistency cannot be pushed down further. As of this moment, the inconsistency crucially uses the Axiom of Replacement. It remains to be seen if the inconsistency can be reworked to attack the earlier system ZC (Zermelo set theory with the axiom of choice). In fact, one expert predicted that the immediate fallback position in the foundations of mathematics will be ZC, and surmised that this will probably – and hopefully – hold. Despite this, he said that there can be no doubt that any confidence that we have in our foundations has been permanently and severely shaken, even if not completely destroyed.

This far more than merely spectacular discovery of Professor Wood is beginning to affect the thinking of mathematicians who work in areas far removed from foundations. Many mathematicians are deeply concerned and want to know if their work is impacted. Specifically, they want to be reassured that their proofs can be cast in so called “safe systems”. Experts in foundations have been generally reassuring them that at this time, all indications are that ZC is safe, and that they have been able to assure all of the mathematicians that have inquired, that their proofs can be done within ZC. However, they cautioned that the confidence in ZFC and much stronger systems has been extremely strong, and if the mathematics community can be so devastatingly wrong about ZFC, then why can’t they be equally wrong about ZC?

One interesting exception to the adequacy of ZC is the highly regarded theorem of Donald A. Martin called Borel determinacy, which was shown by Harvey M. Friedman to not be provable in ZC. Friedman established that Martin’s theorem is, in a precise sense, stronger than ZC but – by Martin – it is weaker than ZFC. Wood is not yet sure if his methods would demolish the relevant extensions of ZC that lie well below ZFC. If so, then Wood would be refuting Martin’s theorem – a shocking blow to this celebrated senior figure (Martin) in foundations.

Wood’s shock has created such excitement at Havad that a press conference was held last week featuring Professor Wood, Professor Koller, and President Faust, followed by an all day meeting led by Wood and Koller. Koller is a Professor of Philosophy here, who specializes in the philosophy of mathematics. The Havad Mathematics and Philosophy Departments regarded this Wood development as of such staggering epic importance that, with the enthusiastic approval of President Faust, they asked all professors in their two departments to cancel all of their classes for a day, and urge all students to attend the meeting. Attendance at the meeting was very strong.

President Faust opened the meeting with a statement. She said that only occasionally has a breakthrough been achieved by Havad faculty that demands immediate special recognition across our entire community. I have urgently convened an ad hoc committee and the Trustees for the immediate appointment of Professor Wood to University Professor. The vote was unanimous after only a few minutes of discussion, which is remarkable given that  Professor Wood has only recently arrived at Havad. We will also be featuring the work of Professor Wood in a special fund raising campaign for the Mathematics and Philosophy Departments. Faust said that her office has contacted many leading scholars across mathematics, science, and philosophy, and they all agree that Professor Wood’s ideas have great promise for future developoments, and promise to have an impact on the history of mathematics and philosophy comparable to that of relativity and quantum mechanics in physics and DNA in biology. At the moment, this impact can be viewed as spectacularly negative and shocking, with a surprise factor arguably greater than the aforementioned revolutions. It is too early to tell what positive developments will come out of the utter destruction of our accepted foundations for mathematics, but the full implications of scientific and philosophical revolutions take time to evolve.

At the meeting, Professor Wood was very understated and cautious, leaving the fireworks to Professor Koller. Wood confined his remarks mostly to the retracing of the insights that led to the inconsistency. He said that while working on his favorite set theoretic problem, the continuum hypothesis (CH), within a framework far stronger than ZFC, he was able to recently resolve some crucial technical questions that had eluded him for many years. He was able to refute certain so called “large large cardinal hypotheses” which he was on record as “looking suspicious”. But then he saw that the core of the argument could be modified to work with weaker and weaker large cardinal hypotheses, all the way down to ZFC itself. At first, Wood thought he was simply making some subtle mistakes and that he had better be more careful so as to not waste any more time. But then he found that there were in fact no errors, and that ZFC itself had been destroyed. Experts in set theory seem to have little trouble following his general outline, and have poured over the detailed manuscript to their satisfaction. However, the rest of the audience was clearly lost at an early stage, but were so mesmerized by the event that they stayed until the very end and had nearly universal expressions of utter fascination and deep respect.

Koller delivered a fascinating heart felt self deprecating presentation to the effect that Wood’s discovery had completely refuted virtually all of his own work in philosophy of mathematics, and that he is in a devastating state of philosophical paralysis. He said he even drafted a resignation letter to his Department chair. But he never sent it. Koller said that it was too early to tell what kind of philosophy of mathematics now makes sense in light of Wood’s revolutionary discovery, and he now wants to help rebuild his own philosophy of mathematics. He says he intends to collaborate with a colleague, Professor Gold, in the philosophy department, also a philosopher of mathematics, who has long been skeptical of a heavily set theoretic approach to the foundations of mathematics. Koller also said that Wood’s recent work utterly destroys the overwhelming majority of Wood’s previous work (with some notable exceptions particularly in functional analysis), and he (Koller) thinks that not even ZC is safe from the likes of Wood. But he is also confident that foundations of mathematics will be successfully rebuilt, and yield unpredictable fruits of a wholly positive nature as an outgrowth of this spectacularly devastating event.

The Press Office has received advanced word that at the suggestion of the American Mathematical Society, the International Mathematical Union is urgently convening, concerning a special award for Professor Wood, as he is no longer eligible for the prestigious Fields Medal. Such a special recognition has only been conferred on Professor Andrew Wiles for his proof of Fermat’s Last Theorem, while he was on the faculty at [our arch rival] Princeton University.​Professor Wood’s epoch shocking discovery may even cast doubt on Wiles’ proof, in that his original proof uses the full power of the demolished ZFC. However, later investigations spearheaded by Colin McLarty have pushed the FLT proof down well within ZC​, and there is hope for pushing the FLT proof down much further. Wood’s breakthrough has greatly stirred interest in determining just what axioms of mathematics are really needed to prove FLT.

Although both the Wiles and Wood developments are very dramatic, ​there can be no comparison between the general intellectual interest and impact of Wood ​over that of Wiles. On this basis, it is transcendentally greater, as it profoundly affects the relationship that many mathematicians and philosophers have with their own ​subjects, at the deepest personal level. Furthermore, it is a truly sensational totally unexpected surprise, coming out of essentially nowhere by a single individual.

Havad Press Office
August 23, 2014

Re: Paper and slides on indefiniteness of CH

Hugh Woodin wrote:

What about impact? I think it is clear that an inconsistency in ZFC+PD would be widely regarded as the greatest theorem in the history of mathematics and would have tremendous intellectual impact. It would certainly generate considerable press.

Perhaps I am too close to PD, but replace ZFC+PD by ZFC in this discussion. It does not really change anything except the impact factor increases.

Very interesting. I will summarize my own view by constructing two PRESS RELEASES, one for ZFC and the other for ZFC + PD.


URGENT PRESS RELEASE
HAVAD UNIVERSITY

Professor Wood, who recently moved to the Havad mathematics department from Berkeley, has stunned the mathematical and philosophical world with his breathtaking demolition of the standard foundations for mathematics that has been almost universally accepted since the 1920’s. In a development of epoch proportions, Wood has actually shown that the usual ZFC axioms for mathematics are in fact inconsistent. For example, Wood has been able to prove from the ZFC axioms that both 2+2 = 4 and 2+2 = 5.

All experts in the foundations of mathematics interviewed considered this development to be astonishing beyond belief, as it threatens to throw the foundations of mathematics into a complete state of utter chaos. They agreed that the only chance for some calm would be if the inconsistency cannot be pushed down further. As of this moment, the inconsistency crucially uses the Axiom of Replacement. It remains to be seen if the inconsistency can be reworked to attack the earlier system ZC (Zermelo set theory with the axiom of choice). In fact, one expert predicted that the immediate fallback position in the foundations of mathematics will be ZC, and surmised that this will probably – and hopefully – hold. Despite this, he said that there can be no doubt that any confidence that we have in our foundations has been permanently and severely shaken, even if not completely destroyed.

This far more than merely spectacular discovery of Professor Wood is beginning to affect the thinking of mathematicians who work in areas far removed from foundations. Many mathematicians are deeply concerned and want to know if their work is impacted. Specifically, they want to be reassured that their proofs can be cast in so called “safe systems”. Experts in foundations have been generally reassuring them that at this time, all indications are that ZC is safe, and that they have been able to assure all of the mathematicians that have inquired, that their proofs can be done within ZC. However, they cautioned that the confidence in ZFC and much stronger systems has been extremely strong, and if the mathematics community can be so devastatingly wrong about ZFC, then why can’t they be equally wrong about ZC?

One interesting exception to the adequacy of ZC is the highly regarded theorem of Donald A. Martin called Borel determinacy, which was shown by Harvey M. Friedman to not be provable in ZC. Friedman established that Martin’s theorem is, in a precise sense, stronger than ZC but – by Martin – it is weaker than ZFC. Wood is not yet sure if his methods would demolish the relevant extensions of ZC that lie well below ZFC. If so, then Wood would be refuting Martin’s theorem – a shocking blow to this celebrated senior figure (Martin) in foundations.

Wood’s shock has created such excitement at Havad that a press conference was held last week featuring Professor Wood, Professor Koller, and President Faust, followed by an all day meeting led by Wood and Koller. Koller is a Professor of Philosophy here, who specializes in the philosophy of mathematics. The Havad Mathematics and Philosophy Departments regarded this Wood development as of such staggering epic importance that, with the enthusiastic approval of President Faust, they asked all professors in their two departments to cancel all of their classes for a day, and urge all students to attend the meeting. Attendance at the meeting was very strong.

President Faust opened the meeting with a statement. She said that only occasionally has a breakthrough been achieved by Havad faculty that demands immediate special recognition across our entire community. I have urgently convened an ad hoc committee and the Trustees for the immediate appointment of Professor Wood to University Professor. The vote was unanimous after only a few minutes of discussion, which is remarkable given that Professor Wood has only recently arrived at Havad. We will also be featuring the work of Professor Wood in a special fund raising campaign for the Mathematics and Philosophy Departments. Faust said that her office has contacted many leading scholars across mathematics, science, and philosophy, and they all agree that Professor Wood’s ideas have great promise for future developoments, and promise to have an impact on the history of mathematics and philosophy comparable to that of relativity and quantum mechanics in physics and DNA in biology. At the moment, this impact can be viewed as spectacularly negative and shocking, with a surprise factor arguably greater than the aforementioned revolutions. It is too early to tell what positive developments will come out of the utter destruction of our accepted foundations for mathematics, but the full implications of scientific and philosophical revolutions take time to evolve.

At the meeting, Professor Wood was very understated and cautious, leaving the fireworks to Professor Koller. Wood confined his remarks mostly to the retracing of the insights that led to the inconsistency. He said that while working on his favorite set theoretic problem, the continuum hypothesis (CH), within a framework far stronger than ZFC, he was able to recently resolve some crucial technical questions that had eluded him for many years. He was able to refute certain so called “large large cardinal hypotheses” which he was on record as “looking suspicious”. But then he saw that the core of the argument could be modified to work with weaker and weaker large cardinal hypotheses, all the way down to ZFC itself. At first, Wood thought he was simply making some subtle mistakes and that he had better be more careful so as to not waste any more time. But then he found that there were in fact no errors, and that ZFC itself had been destroyed. Experts in set theory seem to have little trouble following his general outline, and have poured over the detailed manuscript to their satisfaction. However, the rest of the audience was clearly lost at an early stage, but were so mesmerized by the event that they stayed until the very end and had nearly universal expressions of utter fascination and deep respect.

Koller delivered a fascinating heart felt self-deprecating presentation to the effect that Wood’s discovery had completely refuted virtually all of his own work in philosophy of mathematics, and that he is in a devastating state of philosophical paralysis. He said he even drafted a resignation letter to his Department chair. But he never sent it. Koller said that it was too early to tell what kind of philosophy of mathematics now makes sense in light of Wood’s revolutionary discovery, and he now wants to help rebuild his own philosophy of mathematics. He says he intends to collaborate with a colleague, Professor Gold, in the philosophy department, also a philosopher of mathematics, who has long been skeptical of a heavily set theoretic approach to the foundations of mathematics. Koller also said that Wood’s recent work utterly destroys the overwhelming majority of Wood’s previous work (with some notable exceptions particularly in functional analysis), and he (Koller) thinks that not even ZC is safe from the likes of Wood. But he is also confident that foundations of mathematics will be successfully rebuilt, and yield unpredictable fruits of a wholly positive nature as an outgrowth of this spectacularly devastating event.

The Press Office has received advanced word that at the suggestion of the American Mathematical Society, the International Mathematical Union is urgently convening, concerning a special award for Professor Wood, as he is no longer eligible for the prestigious Fields Medal. Such a special recognition has only been done for Professor Andrew Wiles for his work on Fermat’s Last Theorem, while he was on the faculty at [our arch rival] Princeton University. Although both of these developments are dramatic, there can be no comparison between the general intellectual interest and impact of Wood as opposed to that of Wiles. On this basis, it is transcendentally greater, as it profoundly affects the relationship that many mathematicians and philosophers have with their subjects, at the deepest personal level. Furthermore, it is a truly sensational totally unexpected surprise, coming out of essentially nowhere by a single individual.

Havad Press Office
August 23, 2014


PRESS RELEASE
HAVAD UNIVERSITY

Professor Wood, who recently moved to the Havad mathematics department from Berkeley, has stunned the set theory community with his breathtaking demolition of certain so called large cardinal hypotheses. The demolished large cardinal hypotheses had been long advocated by most set theorists as important additions to the usual ZFC axioms that have been the almost universally accepted foundations for mathematics since the 1920’s. These large cardinal hypotheses were particularly advocated because of their consequences for certain classical problems in an area called higher descriptive set theory.

In (ordinary) descriptive set theory, one studies the structure of Borel measurable sets and functions on complete separable metric spaces, and these are familiar to most mathematicians. By and large, the area does not present any foundational problems, and proceeds as normal mathematics. However, in higher descriptive set theory, Borel measurability is vastly generalized by the so called projective hierarchy of sets, which involves closing off under Boolean operations and images under Borel functions. By prior work of Martin, Steel, and Wood, it was established that virtually all of the main results in descriptive set theory, when lifted to the projective hierarchy, can be settled with certain large cardinal hypotheses. These includes virtually all of the open questions left open in the area by its founders in the first half of the 20th century. It should be noted that the hypothesis “all sets are constructible”, or V = L, was well known to also settle all of these open questions, but V = L is almost universally rejected as a reasonable axiom of set theory by the set theory community.

Wood’s pathbreaking and spectacular work actually refutes what is called projective determinacy. This is the generalization of Martin’s celebrated theorem to the projective sets. Martin proved within the usual ZFC axioms for mathematics, that all Borel measurable sets are “determined”, — a concept from infinite game theory. Projective determinacy, usually written as PD, asserts that all projective sets are likewise “determined”.

By 1990, from work of Martin, Steel, and Woodin, we know that PD is provable from certain large cardinal hypotheses. In light of Wood’s recent refutation of PD, we see that these large cardinal hypotheses have been refuted.

Experts in the area say that this work has had a devastating and profound impact on the history of set theory, and requires us to rethink much of what we have thought about the foundations of set theory.They report that the result is much more devastating than the last time a large cardinal hypothesis was refuted — back in the late 1960s by Ken Kunen. That earlier much stronger hypothesis had not previously led to any detailed associated structural results of the kind that made the much weaker cardinal hypotheses destroyed by Wood so attractive and compelling for most set theorists. The mourning of the loss of PD and the associated large cardinal hypotheses is just beginning, and where it leads is at this time totally unclear. Most experts, however, do not believe that ZFC itself — the almost universally accepted foundations for mathematics throughout the mathematics community — is seriously threatened by this spectacular work of Wood.

Havad Press Office

August 23, 2014
Harvey Friedman reporting.