Edward Nelson claims proof of inconsistency in Peano Arithmetic

It seems important in this context to distinguish between consistency of different systems

Overall, my consistency estimate for PA if anything has gone up in this context. Because if Edward Nelson tries really hard and fails to find a contradiction that's more evidence that there isn't one. How much should it go up by? I'm not sure.

On the other hand, this may very well make my estimate for consistency of ZFC go down. That's because although in this specific case Nelson's approach didn't work, it opens up new avenues into demonstrating inconsistency, and part of me could see this sort of thing being more successfully applied to ZFC (although that also seems unlikely). (Also note that there have been attempts by a much larger set of mathematicians in the last few years to find an inconsistency in ZF and that has essentially failed.)

One thing that this has also done is made me very aware in a visceral fashion that number theory and logic are not the same things even when one is talking about subsystems of PA. I already knew that, but maybe had not fully emotionally processed it as much as I should have as when I saw Nelson's outline, and then the subsequent discussions and had a lot of trouble following the details. The main impact of this is to suggest that when making logic related claims (especially consistency and independence/undecidability issues) I should probably not rate my own expertise as highly as I do. As a working mathematician, I almost certainly have more relevant expertise than a random individual, but I've probably been overestimating how much my expertise matters. In both the cases of ZF/ZFC and the case of PA, reducing my confidence means increasing the chance that they are inconsistent. But, I'm not at all sure by how much this should matter. So maybe this should leave everything alone?

So overall I'd say around .995 consistency for PA and .99 consistency for ZF. Not much change from the old values.

Yes, stupid mistake of mine. Replace velocity by acceleration and position by velocity and the same argument works. The fallacy is in the vagrepunatr bs yvzvgf.

These equations don't make sense dimensionally. Are there supposed to be constants of proportionality that aren't being mentioned?

That is my guess. The simplest way IMO would be replace the g in eq. 1 by a constant c with units of distance^(-1/2). The differential equation becomes r'' = g c r^1/2, which works dimensionally. The nontrivial solution (eq. 4) is correct with an added (c g)^2 in front.

it's not obvious to me that a curve such as he describes exists.

I'm not sure what you mean here. What could be wrong in principle with a curve h = c r^3/2 describing the shape of a dome, even at r = 0?

Jung vs jr fhccbfr gung gur obqvrf pna cnff guebhtu rnpu bgure? Ubj qbrf gur flfgrz orunir nf gvzr cebterffrf?

Yes, I think so too.

Whoops, you're right. I assumed +1/10 meant +1/10th from +1. I had the whole thing backwards.

Gur fhz bs gur gbgny sbeprf ba nyy cnegvpyrf.

sum(n=0 to inf)
{
    [sum(m=0 to n-1) G*(1/2)^n*(1/2)^m/((1/10)^m-(1/10)^n)^2]
    - [sum(m=n-1 to inf) G*(1/2)^n*(1/2)^m/((1/10)^n-(1/10)^m)^2]
} 

vf yrff guna 0 sbe rirel grez va gur bhgre fhz, rira gubhtu lbh pna pnapry bhg nyy gur grezf ol ernpuvat vagb gur vaare fhzf.

I didn't downvote, but if I had it would have been because you seem to be missing the point a bit.

It may well be the case that we live in a discrete universe without infinities (I assume that's what you meant by integer) but this is not the case for Newtonian Mechanics, which asserts that both space and time remain continuous at every scale. Thomas never claimed to have a paradox in actual physics, only in Newtonian mechanics, so what you are saying is irrelevant to his claim, you might was well bring up GR or QM as a solution.

Appealing to the third law also doesn't help, his whole point is that if you do the calculations one way you get one answer and if you do them another way you get another answer, hence the use of the word paradox.

I didn't downvote you, however both your objections were rather besides the point. One doesn't deal with the issue of Newtonian mechanics, and whether they were indeed as obviously flawed from the beginning -- it feels a bit like explaining the paradox of Achilles reaching the turtle by saying we're living in an integer universe. That may be true, but it feels a rather weak dodging-the-issue explanation.

As for the second answer, it doesn't really state where the calculations in the presentation of the paradox are wrong or are missing something. It just says that they are, so it's a non-solution.

Yes, obviously P(respectable mathematician claims a contradiction | a contradiction exists) > P(respectable mathematician claims a contradiction | no contradiction exists), so it has definitely moved my estimate.

Can you roughly quantify it? Are we talking from million-to-one to million-to-one-point-five, or from million-to-one to hundred-to-one?

I'm not sure if this line of debate is a productive one, the issue will be resolved one way or the other by actual mathematicians doing actual maths, not by you and me debating about priors (to put it another way, whatever the answer ends up being, this conversation will have been wasted time in retrospect).

Sorry if I gave you a bad impression: I am not trying to start a debate in any adversarial sense. I am just curious.

Furthermore, not wanting to be unfair to Nelson, but the fact he's working alone on a task most mathematicians consider a waste of time may suggest a substantial ideological axe to grind (what I have heard of him supports this thoery) and sadly it is easier to come up with a fallacious proof for something when you want it to be true.

Of that there's no doubt, but it speaks well of Nelson that he's apparently resisted the temptation toward self-deceipt for decades, openly working on this problem the whole time.

I can say more if you want, but maybe see here and here for an explanation.

Any finite calculation of superexponentiation will be valid. But as I understand it you can't in general in Nelson's formulation prove that superexponentation is well-defined in general.

My understanding is that Nelson doesn't necessarily believe in exponentiation either although his work here doesn't have the direct potential to wreck that.

I know this isn't the main point of your comment but I got carried away looking this stuff up.

In his 1986 book, he defines a predicate exp(x,k,f) which basically means "f is a function defined for i < k and f(i) = x^i". And he defines another predicate epsilon(k) which is "for all x there exists f such that exp(x,k,f)". I think you can give a proof of epsilon(k) in his regime in O(log log k) steps. In standard arithmetic that takes O(1) steps using induction, but O(log log k) is pretty cheap too if you're not going to iterate exponentiation. So that's a sense in which Nelson does believe in exponentiation.

On the other hand I think it's accurate to say that Nelson doesn't believe that exponentiation is total, i.e. that for all n, epsilon(n). Here's an extract from Nelson's "Predicative Arithmetic" (Princeton U Press 1986). Chapter heading: "Is exponentiation total?"

Why are mathematicians so convinced that exponentiation is total (everywhere defined)? Because they believe in the existence of abstract objects called numbers. What is a number? Originally, sequences of tally marks were used to count things. Then positional notation -- the most powerful achievement of mathematics -- was invented. Decimals (i.e. numbers written in positional notation) are simply canonical forms for variable-free terms of arithmetic. It has universally assumed, on the basis of scant evidence, that decimals are the same kind of thing as sequences of tally marks, only expressed in a more practical and efficient notation. This assumption is based on the semantic view of mathematics, in which mathematical expressions, such as decimals and sequences of tally marks, are regarded as denoting abstract objects. But to one who takes a formalist view of mathematics, the subject matter of mathematics is the expressions themselves together with the rules for manipulating them -- nothing more. From this point of view, the invention of positional notation was the creation of a new kind of number.

How is it then that we can continue to think of the numbers as being 0, S0, SS0, SSS0, ...? The relativization scheme of Chapter 5 [heading: "Induction by relativization"] explains this to some extent. But now let us adjoin exponentiation to the symbols of arithmetic. Have we again created a new kind of number? Yes. Let b be a variable-free term of arithmetic. To say that the expression 2^b is just another way of expressing the variable-free term of arithmetic, SS0 SS0 ... SS0 with b occurrences of SS0, is to assume that b denotes something that is also denoted by a sequence of tally marks. (The notorious three dots are a direct carry-over from tally marks.) The situation is worse for expressions of the form 2^2^b -- then we need to assume that 2^b itself denotes something that is also denoted by a sequence of tally marks.

Mathematicians have always operated on the unchallenged assumption that it is possible in principle to express 2^b as a numeral by performing the recursively indicated computations. To say that it is possible in principle is to say that the recursion will terminate in a certain number of steps -- and this use of the word "numbers" can only refer to the primitive notion; the steps are things that are counted by a sequence of tally marks. In what number of steps will the recursion terminate? Why, in somewhat more that 2^b steps. The circularity in the argument is glaringly obvious.

Well, you'd certainly only need finitely many to prove inconsistency.