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Presentation Report on Dr. Coxs Talk
Philip White
Math 4909/27/2013
Sean Cox, an economist turned set theorist, presented to the
class on September 23, 2013.
The title of Dr. Coxs presentation was Undecidable Problems in
Mathematics, and naturally the
talk was about undecidable problems. Sean Cox surveyed
well-known Undecidable problems
from the various areas of mathematics. As you have probably
noticed, the author reviewed in my
first critical review wrote on a similar topic to Dr. Coxs
presentation, the decision problem. The
reason I keep choosing to write about undecidability, is because
I myself have a strong interest in
the topic. The process has been slow, but I hope to one day
understand Godels formal proofs of
incompleteness. Anyhow, let us continue.
To start the lecture off, Dr. Cox reviewed some well-known
provable, yet extremely
difficult, problems of mathematics. Two of the problems he
mentioned were Fermets Last
Theorem and the Poincare Conjecture. The proof of Fermets Last
Theorem alone is books
length and includes references to thousands of pages of
mathematics. After reviewing the proven
problems, Dr. Cox reviewed a separate category of unknown
mathematical problems, albeit the
reason for their unknown status is likely lack of ingenuity on
the part of the mathematicians. The
two obvious examples of problems from this category are the P=NP
problem and the Riemann
Hypothesis. Finally Dr. Cox mentioned a third category of
problem, truly undecidable problems,
as in, problems undecidable notfrom a lack of ingenuity. For
example, Godel coded statements
like, I am consistent, or, I am not consistent, into a formal
first order mathematical language
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and he rigorously proved that these statements were not provable
in such a framework. Dr.
Cox went on to mention a few other undecidable problems.
I found Dr. Coxs presentation to be very informative, not so
much from the information
presented, but from his style of presentation. When presenting,
it is definitely easier if you have
reminder-notes prepared. I have to give a talk this upcoming
Monday, and I will surely use Dr.
Coxs technique of writing down reminder-notes. When it comes to
Dr. Coxs topic, I am
interested in something he brushed over. He mentioned that
something called Presburger
Arithmetic is complete, however I now that Godels incompleteness
theorem uses a language
only slightly larger than Presburger Arithmetic. I am thoroughly
confused as to why such a small
change can change the status of a mathematical language from
incomplete to complete. I hope to
further investigate this topic.
