What a Logical Mess

I’m scared. Some appears to have proven that the Collatz 3n+1 Conjecture is unprovable. The reason that this result scares me is because it implies that the Collatz 3n+1 Conjecture is true, since if there exists a number n for which there is no m with T^(m)(n)=1, then we can clearly disprove it. I don’t think it should be possible to “prove” statements true using this logic. How can it be verified that a statement is true by showing that it cannot be proven? People have given this argument for the Riemann Hypothesis’s truth if it turns out to be formally undecidable for quite a while, but I don’t know how many people think that such logic can really be used to decide a problem.

On a lighter note, I bought myself a wonderful book yesterday. It’s called Counterexamples in Analysis by Bernard Gelbaum and John Olmsted. It basically goes through all sorts of things that can go wrong in analysis. Do similar books exist for other branches of mathematics?


About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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10 Responses to What a Logical Mess

  1. Holy cow! The paper is fairly short and looks to use elementary arguments, so I’ll have to take a peek and see if it really is good.

    • Simon says:

      Only a little under 2 pages are on the Collatz problem. I have never seen a proof of unprovability before, so I don’t know if this is the normal method of proving it. Some things in it make me suspicious, such as the precise meaning of Lemma 2.

  2. Anonymous says:

    With his reasoning one can, basicly, not factor an arbitrary integer fully.

    • Simon says:

      I don’t know about that. He actually has an argument at least. After all, Lemma 1 is correct, and that says quite a lot. But I don’t know if he has actually proven that the Collatz problem cannot be proven true.

  3. Don’t worry. I’ve run across this guy before. He claimed to have a short elementary proof that P =/= NP, which turned out to be faulty. His proof that Collatz is unprovable doesn’t seem to hold water either. If you look at Theorem 2 (the keystone of the argument) he argues that any proof of Collatz would have infinitely many steps. Um, ever heard of induction?
    Another important point: he doesn’t claim that Collatz is undecidable, merely that it’s impossible to prove true. So it could well be false for all we know.
    I know there’s a book out there called Counterexamples in Topology but I’ve never actually seen it.

  4. aibrainy says:

    The proof looks rather bogus to me as well. It seems that, even if all of his first steps are correct (I’m not at all sure of this either, as Lemma 2 looks a bit iffy), all he has proven is that the time required to compute T^(m)(n) for arbitrary m,n is unbounded. The argument for “Theorem” 2 is completely nonsensical: we don’t necessarily need to find (as he claims) T^(m)(n) = 1 for some m in order to show that the iteration of T eventually hits 1 for any given n. Why people even bother writing papers like this is beyond me.

    • Simon says:

      Well, what does it take for a paper to become a preprint on the Mathematics ArXiv? Doesn’t anyone read them first to see if they’re correct?

      • I was just wondering that, too… Just to hammer the last nail into the coffin, the same argument would work even with 3n+1 replaced with n+1, even though that process clearly converges.

  5. Anonymous says:

    Is Counterexamples in Analysis good? Is it basic or does it have some cool examples too?

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