[FOM] Impact of Inconsistency Proofs

[Mikhail Katz]

I think it should be "epic proportions", not "epoch proportions". The NYT 
copyeditor does it again. MK

On Mon, 25 Aug 2014, Harvey Friedman wrote:

> ÿÿ ÿÿ
> This was sent to another email list in response to a statement made there
> asserting that
>
> 1. An inconsistency in ZFC + PD would be the greatest mathematical theorem
> of all time, with substantial press coverage.
> 2. If it was ZFC, then there would be even greater impact.
>
> The the "press releases" below have been edited to be suitable for the FOM.
>
> URGENT PRESS RELEASE
> WESTGATE
> UNIVERSITY
>
> Professor
> ÿÿ ÿÿ
> Rumek, who recently moved to th
> ÿÿe ÿÿ
> Westg
> ÿÿate ÿÿ
> mathematics department fro
> ÿÿm ÿÿ
> Middlegate, 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,Rumek
> ÿÿ ÿÿ
> has actually shown that the usual ZFC axioms for mathematics are in fact
> inconsistent. For example,
> ÿÿRumek
> 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
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> 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 -
> is weaker than ZFC.
> ÿÿ ÿÿ
> Rume
> ÿÿkÿÿ
> is not sure if his methods would demolish the relevant extensions of ZC
> that lie well below ZFC. If so, then
> ÿÿ ÿÿ
> Rumek
> would be refuting Martin's theorem - a shocking blow to this celebrated
> senior figure (Martin) in foundations.
>
> Rumek's
> ÿÿ ÿÿ
> shock has created such excitement at
> ÿÿWestgate
> that a press conference was held last week featuring Professor
> ÿÿ ÿÿ
> Rumek, Professor
> ÿÿ ÿÿ
> Barner, and President
> ÿÿ ÿÿ
> Willard, followed by an all day meeting led by
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> and
> Barner
> ÿÿ. ÿÿ
> Barner
> ÿÿ ÿÿ
> is a Professor of Philosophy here, who specializes in the philosophy of
> mathematics. The
> ÿÿ ÿÿ
> Westgate
> ÿÿ ÿÿ
> Mathematics and Philosophy Departments regarded this
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> development as of such staggering epic importance that, with the
> enthusiastic approval of President
> ÿÿ ÿÿ
> Willard, 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
> ÿÿ ÿÿ
> Willard
> ÿÿ ÿÿ
> opened the meeting with a statement.
> ÿÿ ÿÿ
> He
> ÿÿ ÿÿ
> said that "only occasionally has a breakthrough been achieved by
> ÿÿ ÿÿ
> Westgate
> ÿÿ ÿÿ
> 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
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> to
> ÿÿ ÿÿ
> ÿÿDistinguished
> Professor. The vote was unanimous after only a few minutes of discussion,
> which is all the more remarkable given that Professor
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> has only recently arrived at
> ÿÿ ÿÿ
> Westgate. We will also be featuring the work of Professor
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> in a special fund raising campaign for the Mathematics and Philosophy
> Departments".
> ÿÿ ÿÿ
> Willard
> ÿÿ ÿÿ
> said that her office has contacted many leading scholars across
> mathematics, science, and philosophy, and they all agree that Professor
> ÿÿ ÿÿ
> Rumek'
> ÿÿ
> ideas have great promise for future
> ÿÿdevelopments
> , 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" according to
> President
> ÿÿ ÿÿ
> Willard
> ÿÿ.ÿÿ
>
> .
> At the meeting, Professor
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> was very understated and cautious, leaving the fireworks to Professor
> ÿÿ ÿÿ
> Barner
> ÿÿ. ÿÿ
> Rumek
> ÿÿ ÿÿ
> 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" about 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,
> ÿÿ ÿÿ
> Rumek
> ÿÿ ÿÿ
> 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.
>
> Barner
> ÿÿ ÿÿ
> delivered a fascinating heart felt self deprecating presentation to the
> effect that
> ÿÿ ÿÿ
> Rumek's
> ÿÿ ÿÿ
> 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.
> ÿÿ ÿÿ
> Barner
> ÿÿ ÿÿ
> said that it was too early to tell what kind of philosophy of mathematics
> now makes sense in light of
> ÿÿ ÿÿ
> Rumek's
> ÿÿ ÿÿ
> revolutionary discovery, and he now wants to help rebuild the philosophy of
> mathematics. He says he intends to collaborate with a colleague, Professor
> ÿÿ ÿÿ
> Tadin, in our philosophy department, also a philosopher of mathematics, who
> has long been skeptical of a heavily set theoretic approach to the
> foundations of mathematics.
> ÿÿ ÿÿ
> Barner
> ÿÿ ÿÿ
> ÿÿ
> also said that
> ÿÿ ÿÿ
> Rumek's
> ÿÿ ÿÿ
> recent work utterly destroys the overwhelming majority of
> Rumek's
> ÿÿ ÿÿ
> previous work (with some notable exceptions particularly in functional
> analysis), and he (Barner) thinks that not even ZC is safe from the likes of
> Rumek. 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
> Rumek, as he is no longer eligible for the prestigious Fields Medal. Such a
> special recognition by the IMU has only been previously
> ÿÿ ÿÿ
> conferred on Professor Andrew Wiles for his proof of Fermat's Last Theorem,
> while he was on the faculty of [our arch
> ÿÿ ÿÿ
> rival
> ÿÿ] Eastgate University.ÿÿ
> Professor Rumek'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.
> ÿÿRumek's
> breakthrough has greatly stirred interest in determining just what axioms
> of mathematics are really needed to prove FLT.
>
> Although both
> ÿÿ ÿÿ
> the Wiles and
> ÿÿRumek
> developments are very dramatic,
> ÿÿ ÿÿ
> there can be no comparison between the general intellectual interest and
> impact of
> ÿÿRumek
> over
> ÿÿ ÿÿ
> that of Wiles. On this basis, it is transcendentally greater, as it
> profoundly affects the relationship that many mathematicians and
> philosophers have with their
> ÿÿ ÿÿ
> own
> ÿÿ ÿÿ
> subjects, at the deepest personal level. Furthermore, it is a truly
> sensational and totally unexpected surprise, coming out of the blue by a
> single individual's monumental insights.
>
>
> ÿÿWestgate
> Press Office
> August 23, 2014
> lightly edited 8/25/14
>
> PRESS RELEASE
> ÿÿWESTGATE
> UNIVERSITY
>
> Professor
> ÿÿRumek
> , who recently moved to the
> ÿÿWestgate
> mathematics department from
> ÿÿMiddlegate
> , 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
> ÿÿWoodin
> , 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.
>
> ÿÿRumek'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, normally
> abbreviated 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
> ÿÿRumek'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 its foundations.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
> ÿÿRumek
> 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
> ÿÿRumek
> .
>
> ÿÿWestgate
> Press Office
> August 23, 2014
> lightly edited 8/25/14
>
> Harvey Friedman reporting.
>