See <a href="http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039531" rel="nofollow">http://golem.ph.utexas.edu/category/2011/09/the_inconsistenc...</a> for interesting discussion of this result.<p>However <a href="http://xkcd.com/955/" rel="nofollow">http://xkcd.com/955/</a> applies in spades here. While it would be really interesting if Peano arithmetic is inconsistent, it is also really unlikely. For those who don't know the Peano axioms, see <a href="http://en.wikipedia.org/wiki/Peano_axioms" rel="nofollow">http://en.wikipedia.org/wiki/Peano_axioms</a>.<p>Of those axioms, nobody has any trouble with any axiom other than the last one. Which is induction. His claim boils down to stating that allowing proof by induction leads to contradictions. But there is a pretty big tower of theorems needed to get there. Most people's guess is that he has made a mistake. If he hasn't, then it is still far more likely that some theorem in that tower is wrong than that induction leads to contradictions.<p>However if he proves to be right, in that tiny sliver, this will be really cool.<p>Now for the people who think I'm wrong, anyone want to wager $200 on the outcome? :-)
What's interesting is that proof is computer assisted:<p><i>The proofs are automatically checked by a program I devised called qea (forquod est absurdum, since all the proofs are indirect). Most proof checkers require one to trust that the program is correct, something that is notoriously difficult to verify. But qea, from a very concise input, prints out full proofs that a mathematician can quickly check simply by inspection. To date there are 733 axioms, definitions, and theorems, and qea checked the work in 93 seconds of user time, writing to files 23 megabytes of full proofs that are available from hyperlinks in the book.</i>
For those who are wondering if this is a crank. He is a notable mathematician. He is most notable for his work on <a href="http://en.wikipedia.org/wiki/Internal_set_theory" rel="nofollow">http://en.wikipedia.org/wiki/Internal_set_theory</a> which is a way to simplify the handling of non-standard analysis (calculus with infinitesimals - hyperreals on a rigorous footing).<p>He also wrote this book <a href="http://www.math.princeton.edu/~nelson/books/rept.pdf" rel="nofollow">http://www.math.princeton.edu/~nelson/books/rept.pdf</a> that studies probability theory without measure theory - I've only been through a couple chapters but I recommend it as interesting. Different perspectives help to allow one to understand things more fully.
Sensationalist title. A better one would be "Unreviewed proof claims that Peano arithmetic is inconsistent". This is self-published, hasn't been looked at by anybody else, and contradicts a very well-accepted and thoroughly examined argument for consistency from way back in 1936.<p>1: <a href="http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof" rel="nofollow">http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof</a><p>[edit: as the commenters point out, I'm no mathematician, and I'm working off Wikipedia. Go with what they say.]
Nelson is a full professor of mathematics at Princeton.<p>Some more discussion here: <a href="http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html" rel="nofollow">http://golem.ph.utexas.edu/category/2011/09/the_inconsistenc...</a><p>Notably, Terry Tao doesn't buy it.
Ed Nelson has now withdrawn his claim.<p><a href="http://www.cs.nyu.edu/pipermail/fom/2011-October/015832.html" rel="nofollow">http://www.cs.nyu.edu/pipermail/fom/2011-October/015832.html</a><p><pre><code> Terrence Tao, at
http://golem.ph.utexas.edu/category/2011/09/
and independently Daniel Tausk (private communication)
have found an irreparable error in my outline.
In the Kritchman-Raz proof, there is a low complexity
proof of K(\bar\xi)>\ell if we assume \mu=1, but the
Chaitin machine may find a shorter proof of high
complexity, with no control over how high.
My thanks to Tao and Tausk for spotting this.
I withdraw my claim.
The consistency of P remains an open problem.</code></pre>
If this is true, it would be a bigger deal to logicians than a solution to P vs NP.<p>My guess is that there will end up being a mistake somewhere (based purely on statistics of huge claims without full posted proofs), but it seems too early to say anything definitive. Terence Tao has guessed it is wrong, which is evidence against it, but then Nelson acknowledged Tao's remarks and said it isn't a problem, which is interesting. This is making me wish I was more of an expert so I could decide for myself.
I think the write-up has some bad signs (no central point, lots of side discussion, no new method ... see: <a href="http://www.scottaaronson.com/blog/?p=304" rel="nofollow">http://www.scottaaronson.com/blog/?p=304</a> ).
IANAMathematician, but isn't this implied by the incompleteness theorem? Is this result significant because the seams in the consistency of Peano arithmetic haven't been formalized yet?<p>edit: After a quick visit to Wikipedia, it appears that completeness and consistency are actually separate things. I'll go back to coding now.