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
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
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.