[FOM] Impact of Inonsistency/press releases

Harvey Friedman hmflogic at gmail.com
Sat Aug 30 23:37:18 EDT 2014

I posted this earlier to the FOM. It was composed in gmail, and it was
badly mangled, at least on the FOM Archives. From now on, I will compose
elsewhere and copy into gmail before sending it. This seems to work much
better for me.

So I am resending this so it hopefully be much more readable.

Impact of Inconsistency Proofs

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 "press releases" below have been edited to be suitable for the FOM.


Professor Rumek, who recently moved to th​e ​Westg​ate ​mathematics
department fro​m ​
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 has actually shown that the usual ZFC axioms
for mathematics are in fact
inconsistent. For example, ​Rumek 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 Rumek is
beginning to affect the thinking of mathematicians who work in areas
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.
​ ​
Rume​k​ is not sure if his methods would demolish the relevant extensions of ZC
that lie well below ZFC. If so, then Rumek would be refuting Martin's
theorem - a shocking blow to this celebrated
senior figure (Martin) in foundations. Rumek's shock has created such
excitement at ​Westgate that a press conference was held last week
featuring Professor
​Rumek, Professor Barner, and President Willard, followed by an all
day meeting led by Rumek and Barner. Barner is a Professor of
Philosophy here, who specializes in the philosophy of

The ​Westgate Mathematics and Philosophy Departments regarded this
Rumek development as of such staggering epic importance that, with the
enthusiastic approval of President 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 Willard opened the meeting with a statement. He said that
"only occasionally has a breakthrough been achieved by Westgate
faculty that demands immediatespecial recognition across our entire
community. I have urgently convened an ad hoc committee and the
Trustees for the immediate appointment of Professor Rumek t
Professor. The vote was unanimous after only a few minutes of
discussion, which is all the more remarkable given that Professor
Rumek has only recently arrived at
​Westgate. We will also be featuring the work of Professor Rumek in a
special fund raising campaign for the Mathematics and Philosophy
​ ​
Willard said that her office has contacted many leading scholars across
mathematics, science, and philosophy, and they all agree that Professor
​Rume's ideas have great promise for future developments, 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 Willard.

At the meeting, Professor Rumek was very understated and cautious,
leaving the fireworks to Professor Barner. ​Rumek 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 shich 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, Rumek 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

Barner delivered a fascinating heart felt self deprecating presentation to the
effect that Rumek'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.
​ ​
Barner said that it was too early to tell what kind of philosophy of mathematics
now makes sense in light of Rumek's revolutionary discovery, and he
now wants to help rebuild the philosophy of
mathematics. He says he intends to collaborate with a colleague, Professor
​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.
​ ​
Barner also said that Rumek's recent work utterly destroys the
overwhelming majority of
Rumek's 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 conferred on
Professor Andrew Wiles for his proof of Fermat's Last Theorem,
while he was on the faculty of [our arch rival] Eastgate University.​

Professor Rumek's epic 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. ​Rumek's breakthrough has greatly stirred
interest in determining just what axioms
of mathematics are really needed to prove FLT.

Although both the Wiles and ​Rumek developments are very dramatic,
there can be no comparison between the general intellectual interest
and impact of Rumek 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 and totally unexpected surprise, coming out of the blue by a
single individual's monumental insights.

Westgate Press Office
August 23, 2014
lightly edited 8/25/14


Professor ​Rumek, who recently moved to the Westgate mathematics department from
​Middlegate, 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
​Woodin, it was established that virtually all of the main results in
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.

​Rumek'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

By 1990, from work of Martin, Steel, and Woodin, we know that PD is
provable from certain large cardinal hypotheses. In light of
​Rumek'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
 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

Press Office
August 23, 2014
lightly edited 8/25/14

Harvey Friedman reporting.
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