[FOM] Impact of Inconsistency Proofs
Harvey Friedman
hmflogic at gmail.com
Mon Aug 25 22:08:06 EDT 2014
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
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 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.
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
mathematics. 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 immediate special recognition across our entire
community. I have urgently convened an ad hoc committee and the Trustees
for the immediate appointment of Professor
Rumek
to
Distinguished
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
Departments".
Willard
said that her office has contacted many leading scholars across
mathematics, science, and philosophy, and they all agree that Professor
Rumek'
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 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,
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
respect.
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 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.
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
PRESS RELEASE
WESTGATE
UNIVERSITY
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 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.
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
"determined".
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
by
Rumek
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
Rumek
.
Westgate
Press Office
August 23, 2014
lightly edited 8/25/14
Harvey Friedman reporting.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20140825/99860fc5/attachment-0001.html>
More information about the FOM
mailing list