
Peano arithmetic is probably inconsistent - mike_esspe
http://www.cs.nyu.edu/pipermail/fom/2011-September/015816.html
======
btilly
See
[http://golem.ph.utexas.edu/category/2011/09/the_inconsistenc...](http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039531)
for interesting discussion of this result.

However <http://xkcd.com/955/> 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
<http://en.wikipedia.org/wiki/Peano_axioms>.

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.

However if he proves to be right, in that tiny sliver, this will be really
cool.

Now for the people who think I'm wrong, anyone want to wager $200 on the
outcome? :-)

~~~
lcargill99
Overturning induction is a pretty tall order.

~~~
chernevik
The longest road begins with a single step . . .

~~~
cperciva
... and has the property that if every step is followed by another step, then
the entire road will be walked.

------
mike_esspe
What's interesting is that proof is computer assisted:

 _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._

~~~
joe_the_user
I suspect that the theorem itself would prove the unreliability of the
program. I suppose that puts you in a "bit of a pickle".

~~~
thyrsus
I don't think so; from a quick glance, I believe the theorem deals with
inconsistencies when dealing with infinities, and the program and its output
are all finite. If the program hangs or runs out of disk space, then that
might be confirming evidence for the theorem ;-).

~~~
joe_the_user
No,

An consistent theorem system can know no bounds. There's halfway. If Peano
arithmetic is inconsistent every statement in it is provably true and false.
QED.

------
Dn_Ab
For those who are wondering if this is a crank. He is a notable mathematician.
He is most notable for his work on
<http://en.wikipedia.org/wiki/Internal_set_theory> which is a way to simplify
the handling of non-standard analysis (calculus with infinitesimals -
hyperreals on a rigorous footing).

He also wrote this book <http://www.math.princeton.edu/~nelson/books/rept.pdf>
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.

~~~
mturmon
Radically Elementary Probability Theory (REPT above) is an excellent book. A
very thin volume that anyone who labored through measure-theoretic probability
would enjoy at least scanning.

It is fascinating and original, like the proof mechanism mentioned upthread.

------
saulrh
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.

1: <http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof>

[edit: as the commenters point out, I'm no mathematician, and I'm working off
Wikipedia. Go with what they say.]

~~~
janjan
For this kind of stuff there should be a way to flag a submission as
'sensationalist title' or 'likely false'. I just don't have the time to read
all the comments for all submission just to find out if the submission is true
or not.

~~~
ionfish
Then don't read them; the community will figure out over the next few months
whether Nelson's proved what he's claimed. The Peano axioms have stood for 120
years, it's probably not imperative that you find out whether they've been
shown to be inconsistent right this minute.

~~~
ramchip
> Then don't read them

I think that's his point: it's difficult to differentiate things that are
likely wrong/false and things that have been verified by the community. Flags
could help that.

I prefer simply relying on the comments, personally. In the comments you can
see who's knowledgeable about the topic and often they'll explain what's wrong
with the article. Votes and flagging can't do that.

~~~
ionfish
Flags are a binary mechanism just like voting, and as such are vulnerable to
precisely the same epistemic problems. As you say, comments offer a way of
demonstrating expertise, not merely asserting it.

For this reason someone interested in a disputed topic such as this will
either have to read the comments, in the hope of discovering the main lines of
argument and improving the likelihood that the picture they form is correct,
or suspend their judgement until such a time as better evidence (in this case,
peer review of the claimed proof) is available.

------
billjings
Nelson is a full professor of mathematics at Princeton.

Some more discussion here:
[http://golem.ph.utexas.edu/category/2011/09/the_inconsistenc...](http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html)

Notably, Terry Tao doesn't buy it.

~~~
Steko
Nelson's short reply to one of Tao's comments:

[http://www.cs.nyu.edu/pipermail/fom/2011-September/015826.ht...](http://www.cs.nyu.edu/pipermail/fom/2011-September/015826.html)

------
ionfish
Ed Nelson has now withdrawn his claim.

<http://www.cs.nyu.edu/pipermail/fom/2011-October/015832.html>

    
    
        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.

------
tylerneylon
If this is true, it would be a bigger deal to logicians than a solution to P
vs NP.

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.

------
jmount
I think the write-up has some bad signs (no central point, lots of side
discussion, no new method ... see: <http://www.scottaaronson.com/blog/?p=304>
).

------
jallmann
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?

edit: After a quick visit to Wikipedia, it appears that completeness and
consistency are actually separate things. I'll go back to coding now.

~~~
ionfish
One consequence of the second incompleteness theorem is that no consistent
arithmetic theory of sufficient strength can express its own consistency. One
way of proving such a theory inconsistent is therefore to find a proof within
that theory of its own consistency.

A complete theory is one in which, for any statement φ in the language of that
theory, either φ is provable or ¬φ is provable within that theory. Note that
this is a different sense of completeness than that proven in Gödel's
Completeness Theorem, which states that any sentence satisfied by all models
of a theory is provable.

A consistent theory is one which contains no contradictions. Because
mathematics generally employs classical logic it is explosive [1] and any
contradiction allows one to derive any sentence whatsoever in the language of
the theory as a theorem. Because of this an alternative way to say that a
theory is inconsistent is to say that all the sentences in the language of the
theory are theorems.

[1] <http://en.wikipedia.org/wiki/Principle_of_explosion>

