Talk: Fusible number

It turns out that some of the results of Jeff Erickson have been overturned by Junyan Xu, namely a recursive formula given by Erickson turns out to be wrong. One result of this is that -log_2 m(3) is not 1541023937, but is in fact greater than 2↑↑↑↑↑↑↑↑↑16! Deedlit11 (talk) 15:57, November 16, 2012 (UTC)154102393715410239371541023937

Holy mother of God! We seem to have underestimated the power of the fuse. Good find, Deedlit. FB100Z • talk • contribs 06:26, November 17, 2012 (UTC)

I don't think there is any real evidence that f(x) is on the order of \(F_{\epsilon_0}\). What is believed (but not yet proven, I think) is that the fusible numbers have order type \(\epsilon_0\), but that is completely different. Will amend the article. Deedlit11 (talk) 15:56, February 28, 2013 (UTC)

Proof that fusible numbers are well-ordered, and their order type is at least \(\epsilon_0\). To be exact - there exists strict subset of fusible numbers with order type \(\epsilon_0\), and, from order type definition, we can't embed larger ordinal into smaller ordinal in order-preserving way. I like fact that this implies that this well order can't be proven in PA. LittlePeng9 (talk) 17:46, February 28, 2013 (UTC)

Hmmm, does the fact that fusible numbers have order type at least \(\epsilon_0\) imply that PA cannot prove the fusible numbers are well-ordered?  I have to think about it.  If so, it is a remarkable fact!  There aren't all that many simple independence results from PA, and this would be one of them.  What is especially surprising to me is that Junyan's proof of well-ordering in the paper seems so simple.  Is there a known independence result with such a simple proof?

Unfortunately, this has nothing to do with the growth rate of f(x).  For example, it is easy enough to embed a set of order type \(\epsilon_0\) in the real line such that it number inhabits [n, n+1/2) for any integer n.  This would imply that f(n), as defined in the main article, would be <= 1 for all integers n!  Obviously this is not the case, but it illustrates that the order type of a set and the f(n) defined in the article have no relation. 

Of course, we can guess that f(n) is quite fast-growing, on the basis of f(3) alone! Deedlit11 (talk) 18:19, February 28, 2013 (UTC)

Well, PA can prove that order type of fusible numbers is at least \(\epsilon_0\), but can't prove that it is well order. In PA \(\epsilon_0\) is definable sort of like a concept, but not well founded ordinal. If it could prove well ordering of fusible numbers, it could prove that equivalent ordinal is well founded, and thus \(\epsilon_0\) is well founded and well ordered. Conclusion - well order of fusible numbers is unprovable in PA. But still, remember that this simple proof uses Dependent Choice.

I also wanted to point out that, for any \(\epsilon >0\) there is n such that for any real number \(x>n\) we have \(f(x)<\epsilon \). Informally, set of fusible numbers gets denser and denser as we approach infinity. LittlePeng9 (talk) 19:05, March 1, 2013 (UTC)

Do you mean \(m(x) < \epsilon\)? FB100Z • talk • contribs 23:21, March 1, 2013 (UTC)

Yes, my mistake. LittlePeng9 (talk) 07:32, March 2, 2013 (UTC)

We created the list of functions, and it is necessary to place fusible margin function anyways. Maybe we can state that it has order type, say, at least \(\omega^\omega\). Ikosarakt1 (talk ^ contribs) 18:07, March 18, 2013 (UTC)

If we don't know, we can say we don't know. FB100Z • talk • contribs 18:15, March 18, 2013 (UTC)

Computability

It's not clear to me whether \(m(x)\) is a computable function, although I wouldn't be surprised if it is. Anyone know? you're.so.pretty! 19:01, July 6, 2014 (UTC)

It is. I don't know how it exactly works, but you can find Mathematica algorithm in "Survey on fusible numbers" paper. LittlePeng9 (talk) 20:02, July 6, 2014 (UTC)

I don't trust that paper after Junyan Xu showed that some of Erickson's results are incorrect... you're.so.pretty! 20:14, July 6, 2014 (UTC)

"Survey on fusible numbers" is Xu's paper. LittlePeng9 (talk) 20:24, July 6, 2014 (UTC)

File:Sculpture facepalm.jpg

you're.so.pretty! 20:30, July 6, 2014 (UTC)

It's possible to define some sort of algorithm which computes $\Sigma(n)$, the only thing is that such an algorithm would require infinite time to terminate. But what if $m(x)$ requires infinite time too? Ikosarakt1 (talk ^ contribs) 20:08, July 6, 2014 (UTC)

The definition of 'algorithm' that I am familiar with operates only in finite time frames. you're.so.pretty! 20:14, July 6, 2014 (UTC)

Thanks to well-orderedness of fusibles there most certainly exists finite algorithm for m(x). LittlePeng9 (talk) 20:24, July 6, 2014 (UTC)

I'll believe that for m(x) since there is an algorithm for it in the above paper, but in general, is it obvious that there is an algorithm for the first member of a well-ordered set larger than x? Deedlit11 (talk) 19:03, July 7, 2014 (UTC)

No, it's impossible to decide if, say, number 1/2 is in a given computable well-ordering (corollary of Rice's theorem). However, for sets built in stages, like fusibles, there might be a general way of computing such functions. LittlePeng9 (talk) 19:26, July 7, 2014 (UTC)

This article is due for updates

Erickson and Xu released a paper last year which gives a lot of new information about fusible numbers: https://arxiv.org/abs/2003.14342 I could use some help updating this article appropriately. Otherwise, I'll get to it myself in a few days or something. QuasarBooster (talk) 03:39, 13 January 2021 (UTC)