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Faster-than-light neutrinos? Boring... let's see something really revolutionary. Edward Nelson, a math professor at Princeton, is writing a book called Elements in which he claims to prove the inconsistency of arithmetic. He writes:

"I am writing up a proof that Peano arithmetic (P), and even a small fragment of primitive-recursive arithmetic (PRA), are inconsistent. This is posted as a Work in Progress at

A short outline of the book is at:

The outline begins with a formalist critique of finitism, making the case that there are tacit infinitary assumptions underlying finitism. Then the outline describes how inconsistency will be proved. It concludes with remarks on how to do modern mathematics within a consistent theory."

Thanks to +David Roberts and +Andres Caicedo for pointing this out.

I have no idea if Nelson's proof is correct! He has, however, done good mathematics in the past: in his PhD thesis he was the first to rigorously construct an interacting quantum field theory.﻿
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As you probably know, if you want the question of the consistency of arithmetic to be definitively settled, then you have to root for inconsistency, since Goedel essentially says that you can only prove arithmetic is consistent if it's inconsistent (in which case you can prove anything).

I'm glossing over some nuances here, of course...﻿
His first sentence: "The aim of this work is to show that contemporary mathematics, including Peano arithmetic, is inconsistent, to construct firm foundations for mathematics, and to begin building on those foundations" reminds me of Godel's Theorum. Is it a fool's quest to try to make an consistent arithmetic?﻿
Most logicians don't think the problem is "making a consistent arithmetic" - unlike Nelson, they believe they arithmetic we have now is already consistent. The problem is making a consistent system of arithmetic that can prove itself consistent.

Goedel's theorem says that given certain technical conditions, any system of arithmetic that can prove itself consistent must be inconsistent.

So, the only way out is to develop a system of arithmetic that doesn't obey those 'certain technical conditions'. And since Nelson is no fool, this is what he's trying to do.

One of those 'certain technical conditions' is the idea that the numbers 0, 1, 2, 3, 4, ... obey the principle of mathematical induction. Namely, if

A) some property holds for the number 0,


B) given that this property holds for some number n, it holds for the next number n+1,

then it holds for all numbers. Nelson doubts the principle of mathematical induction, for reasons he explains in his book, so I'm sure his new system will eliminate or modify this principle.

Needless to say, this is a radical step. But vastly more radical is his claim that he can prove ordinary arithmetic is inconsistent. Almost no mathematicians believe that. I bet he's making a mistake somewhere, but if he's right he'll achieve eternal glory.﻿
Aha, I see. Proving that the number line is inconsistent would be quite an achievement.

I've come to believe (no mathematician me) that the number line is consistent only in this universe, and that it becomes inconsistent when you move to a different universe. ANd that makes it very handy because if you want to know if you are in THIS universe here, or in some parallel universe with an alternative you, all you have to do is to count. If it goes 1, 2, 3, 4, 5 and so on, then you are here.﻿
Does anyone have any idea of how long it ought to take for a consensus to form around Nelson's claim? I've looked over his supposed proof, and it seems reasonably short and well-written -- almost certainly a faster read than Deolalikar's claimed P != NP proof last year. But then I'm not a logician or model theorist, and I have no idea how subtle those proofs generally are.﻿

I can never even remember to pinch myself when I'm dreaming!﻿
I have read through the outline. Even though it is too sketchy to count as a full proof, I think I can reconstruct enough of the argument to figure out where the error in reasoning is going to be. Basically, in order for Chaitin's theorem (10) to hold, the Kolmogorov complexity of the consistent theory T has to be less than l. But when one arithmetises (10) at a given rank and level on page 5, the complexity of the associated theory will depend on the complexity of that rank and level; because there are going to be more than 2^l ranks and levels involved in the iterative argument, at some point the complexity must exceed l, at which point Chaitin's theorem cannot be arithmetised for this value of l.

(One can try to outrun this issue by arithmetising using the full strength of Q_0^*, rather than a restricted version of this language in which the rank and level are bounded; but then one would need the consistency of Q_0^* to be provable inside Q_0^*, which is not possible by the second incompleteness theorem.)

I suppose it is possible that this obstruction could be evaded by a suitably clever trick, but personally I think that the FTL neutrino confirmation will arrive first.﻿
I don't think I've seen this level of discussion often on facebook. (: Thanks gentlemen - a fascinating eavesdrop!﻿
+John Baez: Could you point me to a PDF file of his PhD thesis? I tried searching his publication list, but I didn't find the PhD thesis :(﻿
+Ulrik Günther - I don't know if Nelson's thesis is available anywhere, but the result I'm talking about, which I believe was proved in his thesis, appears in this paper:

Edward Nelson, A quartic interaction in two dimensions, in Proceedings of the Conference on the Mathematical Theory of Elementary Particles, held at Endicott House in Dedham, Mass. September 12–15, 1965, ed. by Roe Goodman and Irving Segal, MIT Press, Cambridge, MA, 1966, pp. 69–73.

This is from the days when papers and books were on paper, not PDF files. You can read a discussion of his paper here:﻿
Okay, thanks for the information. I'll try to find it in our university's library :)﻿
You're in the next one over.﻿
+David Bernier wrote: "The part of the outline I really really don't understand is everything to do with the formal systems Q_0 , Q_0^* (roughly) of Robinson Arithmetic (said to be weaker than first-order
Peano Arithmetic)."

If you understand this basic stuff please forgive me for taking you literally:

The system Q is "Robinson arithmetic", a version of arithmetic much weaker than Peano arithmetic because it's finitely axiomatizable and lacks the axiom schema of mathematical induction... but still strong enough that Goedel's theorem applies! There's a nice introduction here:

On the other hand Q_0 is a slight variant of Q, apparently introduced by Nelson. In this variant Nelson's axiom 3 (see above link) is replaced by introducing a unary function symbol P (predecessor) and two axioms:

P0 = 0

not(x = 0) -> x = SPx

The technical advantage is that now no axioms have quantifiers! Goedel's theorem still applies.

I don't understand Q_0^*. The star stands for some kind of "relativization" process I never learned about.﻿
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It's because the n-category cafe has transformed into a portal through which inconsistencies in arithmetic are leaking out into the internet...﻿
Yep. The problem was I changed an epsilon_0 in someone's comment to &epsilon<sub>0</sub> instead of &epsilon;<sub>0</sub>, and then went to sleep instead of checking my work.

Edward Nelson is not folding under Terence Tao's friendly attack.﻿
Btw, +David Roberts, I told Greg about Nelson's new work. But he's probably not thinking about that - he's spending most of his time developing physics in a 4d Riemannian universe for the purposes of a trilogy of novels, of which the first is out: The Clockwork Rocket. Most recently he's been taking advantage of the fact that the Dirac equation has a nice quaternionic formulation in a world with 4 space dimensions and no time:﻿
I'm a big fan - currently halfway through 'Hot Rock' in 'Oceanic'.﻿
I forget how 'Hot Rock' goes... but I've read pretty much all his stuff.﻿
It's about an orphaned planet which was dislodged from its Sun, and spent the next few billion years hurtling through space alone. Yet, mysteriously, its surface is still warm, and indeed supports simple life. This turns out to because... [that's as far as I've got].﻿
Gee, I don't think I've read this one! Don't give it away.﻿
+Terence Tao "but personally I think that the FTL neutrino confirmation will arrive first" -- well, yeah....﻿
But Nelson had the coolest sentence in the copyright page of his book: "All numbers in this book are fictitious, and any resemblance to actual numbers, even or odd, is entirely coincidental."﻿
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