Timeline for What's the order type of the following set?
Current License: CC BY-SA 4.0
13 events
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| May 23 at 0:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 24 at 0:05 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 26, 2023 at 0:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 26, 2023 at 23:34 | answer | added | C7X | timeline score: 1 | |
| Nov 5, 2022 at 1:46 | comment | added | Fedor Pakhomov | @C7X Thank you! Unfortunately I don't know Japanese. I guess, at some point I'll try to figure out what is the construction here looking at the program code and examples. | |
| Nov 5, 2022 at 0:00 | comment | added | C7X | For ease of use, here is an online calculator for finding the nth expansion of a given array (arrays are called matrices colloquially, input with rows enclosed in parens as in the PDF), along with some explanation of the program here. Colloquial discussion borrows some terminology from the worm game, such as finding the "good" and "bad" parts of a matrix, as Beklemishev described the "good" and "bad" parts of a worm. (No ethical implications of such a string are implied by this terminology, I hope...) | |
| Nov 4, 2022 at 23:49 | comment | added | C7X | For inputs of 1 row, the patterns of resemblance connection is strongest: 1-row input gives the almost exact behavior of Kirby-Paris's hydra game, which in turn matches with the behavior of pure order-1 patterns quite well (Wilken has a talk that brings up these kinds of forests for pure order-1 patterns, unfortunately a recording of the blackboard during this segment is lost.) | |
| Nov 4, 2022 at 23:49 | comment | added | C7X | Additionally, soon after a PDF (not peer-reviewed) was written up of a proof that the 2018 patch of the program terminates on inputs [[a,b],[c,d],[e,f],...] of two rows, using an order-preserving injection from this program's "hydra game" to Buchholz's 1984 ordinal notation for $\Pi_1^1-CA_0+BI$. The full program starts with input [[0,0,...,0,0],[1,1,...,1,1]] of length 10, so this is some progress. This PDF uses the notation (a,b)(c,d)... for [[a,b],[c,d],...], and it's in Japanese, I'm not sure if there's a translation. | |
| Nov 4, 2022 at 23:25 | comment | added | C7X | @FedorPakhomov Here's a link to the 2018 patch of the program (the one most interested in this series of programs agree to be the canonical one), written in a dialect of BASIC. A word of warning: almost everything from the original entry to the 2018 version is conjectural, around 2016 it was found that the original 2014 program doesn't terminate on some inputs: it actually computed a non-well-founded notation whose well-founded part is at most $\Gamma_{\omega+1}$. No evidence of nontermination is yet shown for the 2018 version. | |
| Nov 4, 2022 at 14:51 | comment | added | Fedor Pakhomov | @C7X Is there some link to this program? It would be curious to take a look. | |
| Nov 3, 2022 at 2:19 | comment | added | C7X | @FedorPakhomov The story behind this question is interesting: in 2014 there was a contest to name large numbers where an anonymous contestant submitted a program (patched in 2018) claimed to take longer to terminate than Buchholz's hydra game, but its termination was only conjectural. Over the past couple years various amateur enthusiasts (including OP) have tried proving its termination and determining the size of its output, earlier this year OP made the connection that just by coincidence its algorithm looks similar to a variant of the patterns of resemblance comparison algorithm. | |
| Oct 25, 2022 at 10:12 | comment | added | Fedor Pakhomov | This seems to be very much related to the calculation of the order type of Carlson's patterns of resemblance. And the latter question is open since the patterns have been introduced around 20 years. | |
| Oct 25, 2022 at 2:27 | history | asked | Reflecting_Ordinal | CC BY-SA 4.0 |