Talk:Interesting number paradox

message: The numbers in the set are defined as uninteresting. Redefining a number in the set as interesting cannot be done since a set is specifically defined. Saying that an uninteresting number is interesting because it is the smallest number of an uninteresting set is to say the set is improperly defined, making the paradox moot. The number is uninteresting by definition. Saying the number is "smallest" and therefore "interesting" is inconsequential.  — Preceding unsigned comment added by 184.10.252.162 (talk) 05:50, 28 November 2013 (UTC)
replies: text: The point is that the set cannot be well-defined. Although one could comment that if A is the smallest member of the set, which makes it interesting and therefore not in the set, where does that leave A? It's no longer interesting because its reason for being interesting is now invalid. -mattbuck (Talk) 08:56, 28 November 2013 (UTC)
replies: text: That goes against the definition of a set. A set, by definition, is well defined. It is impossible to have an interesting number in an uninteresting set, if and only if there is a defined set of interesting numbers outside of the uninteresting set. Therefore, saying a number is interesting therein is fallacious. However, even if there is a subset of interesting numbers the fallacy is in the assumption that there is a set outside of the uninteresting set with the same definition. Again, this defeats the purpose and disregards the definition of a set. Merely saying a number is interesting is not enough, once it has been defined it is done; such is the definition. The paradox only comes in if you allow yourself to distort the definition of the set for the purpose of the paradox, which is again, a fallacy.  — Preceding unsigned comment added by 184.10.252.162 (talk) 17:11, 30 November 2013 (UTC)
thread_title: The set is defined as uninteresting

message: Both interest and uninteresting is not a paradox. it is like being bored of something exciting, or sad that you'll never be this happy again. Emotions cause their opposites all the time, so there is no paradox.  —Preceding unsigned comment added by 76.204.97.52 (talk) 05:38, 6 December 2008 (UTC)
thread_title: Interest is an emotion, not a number

message: Removed:
:This paradox is, however, resolvable.  It may be noted that in this example 38 is the infimum for the set of dull numbers: while not dull itself, there are no dull numbers less than 38.
This is not a resolution of paradox. It is a treachery, done by substituting the definition. What kind of logical fallacy is this? Mikkalai 23:56, 26 Oct 2004 (UTC)
thread_title:  Resolvable? 

message: (text moved here from Wikipedia:Cleanup/October)
*Interesting number paradox a mathematician has to rewrite this article away from its sensationalist, rather than encyclopedic style. Some statements are plain false if under scrutiny (I've already removed the most evident one). Mikkalai 00:03, 27 Oct 2004 (UTC)
thread_title:  Cleanup notes 

message: This page is written as if the Interesting number paradox is a serious mathematical concept which actually proves something. It's meant to be fun. People reading this artcile will get the impression that there are no uniteresting numbers and that this has been proven mathematically!--Heathcliff 23:35, 17 May 2005 (UTC)
replies: text: Every interesting encyclopaedia contains at least one joke. Otherwise a given encyclopaedia is uninteresting. (That is notwithstanding that it might be the smallest uninteresting encyclopaedia and all the ensuing logical consequences inconvenient to this line of argument given the immediate neighbourhood of content with respect to the family of all interest-driven finite random walks.) --125.253.44.20 (talk) 13:35, 21 October 2013 (UTC)
thread_title: Why is this page so serious?

message: Note: added tone template--Keerllston 11:43, 8 December 2007 (UTC)
replies: text: Every interesting encyclopaedia contains at least one joke. Otherwise a given encyclopaedia is uninteresting. (That is notwithstanding that it might be the smallest uninteresting encyclopaedia and all the ensuing logical consequences inconvenient to this line of argument given the immediate neighbourhood of content with respect to the family of all interest-driven finite random walks.) --125.253.44.20 (talk) 13:35, 21 October 2013 (UTC)
thread_title: Encyclopedias are not joke books

message: Similarly, I removed the mention of surreal numbers.  I don't see how they could have anything to do with the interesting number paradox.  There are sets smaller than the surreals that are not well-ordered. Isomorphic 22:48, 13 November 2005 (UTC)
thread_title: Surreal numbers

message: because it is one of the very few real numbers that have been used throughout history to illustrate the concept of an uninteresting real number. Dmharvey 03:17, 12 March 2006 (UTC)
replies: text: I think the problem here is that any number given as an example of an uninteresting number would in itself be interesting, by virtue of being singled out as an example of an uninteresting number. When I read 197.3341 as an example of an uninteresting number, I immediately assumed it was the author's intent that this, in itself, be a paradox. BGreeNZ 04:28, 11 April 2006 (UTC)
thread_title:  197.3341 is a singularly bad example of an uninteresting real number 

message: I'm trying to track down the source of this paradox, but I haven't had any luck yet. Chris Caldwell traces it back to a letter of G. G. Berry (0000206 in the Russell Archives of McMaster University) and calls it Berry's paradox: "You will often find Berry's paradox stated as 'every integer is interesting.'".  I think Berry's paradox is usually understood to be the proof about the smallest number definable in less than X words/letters/keystrokes, though, and from a quick Google search it appears the letter is about this paradox.
The article suggests that Gardner may be the origin of this paradox.  Any thoughts? CRGreathouse (t | c) 23:29, 16 September 2006 (UTC)
replies: text: I read it in a Swedish book at least 20 years ago. I don't know it if was a Swedish original or an translation. The paradox was that there are not an least interesting integer. The book was about popular mathematics and if I remember correctly that story was about 2 mathematicians that made a cab ride and the number of the cab was dull. I just found that number, its 1729 as mentioned in this discussion page http://en.wikipedia.org/wiki/1729_%28number%29 . I guess that the paradox was written in association with that story. —The preceding unsigned comment was added by 81.231.32.208 (talk) 15:44, 21 April 2007 (UTC)
thread_title:  Origin 

message: I just stumbled onto this page, and as far as I can see, there's nothing the matter with its tone.  I therefore removed the template.  --Slashme (talk) 06:46, 17 July 2008 (UTC)
thread_title:  Tone 

message: The article says:
:One proposed resolution of the paradox asserts that only the first uninteresting number is made interesting by that fact. For example, if 39 and 41 were the first two uninteresting numbers, then 39 would become interesting as a result, but 41 would not since it is not the first uninteresting number. However, this resolution is invalid, since the paradox is proved by contradiction: assuming that there is any uninteresting number, we arrive to the fact that that same number is interesting, hence no number can be uninteresting
Is this refutation original research? It just reiterates the original argument, so in that sense it isn't, but what I mean to ask is whether this is commonly considered to be "the" refutation of the proposed resolution. It is awfully uncharitable to assume that whoever proposed the resolution did not understand the paradox at all!
A more charitable way to interpret the resolution is that it rejects the premise that the least uninteresting number is interesting. Let us call the broadest sense of interesting "ordinarily-interesting". There is another sense, the obviously-interesting, which is something like "is a prime number or ...", which is the starting point. Then there is the meta-interesting, which is "any number that is obviously-interesting or is the least number that is not obviously-interesting". Then you analogously have meta-meta-interesting, and more generally metan-interesting. The resolution, as I see it, says that any meta-interesting number is ordinarily-interesting, but that the same may not hold for metan-interesting numbers with n > 1. Further, for none of these x-interestings does it automatically hold that a number is x-interesting if it is the least number that is not x-interesting, so the original argument fails in this more fine-grained model of interestingness. The resolution simply says that this is a more realistic model. -- Coffee2theorems (talk) 17:04, 12 February 2010 (UTC)
replies: text: It relies on logical induction anyway, even not the more general mathematical induction. For to prove that if p and f(p), f being "the next uninteresting number", one must define f, which this does not do; indeed cannot without resorting to a higher-order logic where we can define "uninteresting", which is of course the entire point. Give that p is uninteresting, f(p) must also be uninteresting, or rather, some function of uninteresting (such as negation), but it doesn't define that function or attempt to: it might be "f(p) = slightly less interesting(p)" or f(p) = "a cow on a bicycle". Si Trew (talk) 23:26, 9 April 2010 (UTC)
thread_title:  Refutation of a proposed resolution in the article 

message: well what is it - 11630 12067  or 12407? Kittybrewster  ☎ 00:14, 13 November 2011 (UTC)
thread_title: Smallest uninteresting number

message: As mentioned here, 12407 does appear in the On-Line Encyclopedia of Integer Sequences, at least implicitly, as it is a semiprime, i.e. 12407 is the product of two prime numbers.  See :oeis:A001358.  This refutes the claim made on the main page.  --Craw-daddy | T | 11:49, 14 November 2011 (UTC)
thread_title: 12407

message: The citation for the claim that 12407 is the first interesting number is taken from Nathaniel Johnson's Blog.  Is this really a suitable source to be referenced in a Wikipedia article?  For reference: Johnston is according to the blog a PhD student. --Colapeninsula (talk) 16:37, 14 November 2011 (UTC)
thread_title:  Nathaniel Johnson Blog a reliable source? 

message: I'm sorry, but if the smallest uninteresting number is interesting, then it's no longer the smallest uninteresting number, and since there was nothing else interesting about it, that means it's not interesting.  Which makes it the smallest uninteresting number again, which makes it interesting again.  Which makes it not interesting, which makes it interesting, which makes it not...  (Note: the previous probably counts as original research, so don't add it to the article.)  Xtifr tälk 10:34, 10 April 2012 (UTC)
replies: text: No, it's a legitimate proof strategy.  See also Euclid's theorem for an analogous proof that there are infinitely many prime numbers.  For more background, see Proof by contradiction.  2620:0:1000:3002:BAAC:6FFF:FE91:91CB (talk) 19:23, 3 January 2013 (UTC)
thread_title:  Obvious flaw 

message: There are useful and therefore interesting numbers in any countable set. Take statistical measures for example, the median, mean, max, and don't forget min, are the most studied and useful numbers. Saying that these numbers are not interesting is like saying that sex isn't interesting. If you don't find sex interesting you are either mentally handicapped (and therefore inhumane) or otherwise lying (which is the likely case).  — Preceding unsigned comment added by 71.220.59.235 (talk) 04:25, 20 October 2013 (UTC)
thread_title:  There are objectively interesting aspects of countable sets. 

message: James Gleick mentions this issue in chapter 12 of his 2010 book "The information".  If we define 'interesting' as any number X being representable by a computer program that can be encoded in less information than X (see Turing numbers), then the first interesting number becomes the first number which cannot be encoded in such a way.  A small modification to the programming language (the addition of the function UNINT(x) may then allow this idea to be encoded in less information than X.  
Unlike the article's uninteresting property, this has built in limits - presumably it cannot be extended indefinitely without making the program encoding expand (due to the addition of new symbols or functions) so that that the new encoding for f(X) exceeds the size of X.  This obviously parallels Shannon's statements about the limits of lossless compression.
Have added the book to the 'further reading' section.  — Preceding unsigned comment added by 69.181.137.62 (talk) 18:59, 25 October 2013 (UTC)
thread_title: Consequences for computer science

message: What proof is there that a "smallest uninteresting number" exists? If you pedantically insist on only looking at natural numbers, then sure; but uninteresting fractions and negative numbers would surely exist, too. 131.191.13.63 (talk) 02:31, 24 December 2013 (UTC)
replies: text: There isn't a proof that such a number exists, that it does is taken as a point of contradiction in the proof - you assume it exists, find it leads you to a contradiction and thus prove it does not in fact exist. You could apply the same logic to the positive real numbers as they are well-ordered (exactly one of a>b, a=b, a(Talk)</small> 09:53, 24 December 2013 (UTC)
replies: text: This section is somewhat off-topic, as the article specifically states that it refers to natural numbers. However, I'm not at all clear what Mattbuck means by "For real numbers it's fine", since there is no guarantee that a subset of the real numbers must have a least element. You can't just pick two real numbers (such as 1 & -1) and say that doing so is done "without loss of generality", because you are assuming that there are two numbers in the set with minimal absolute value, which is a substantial loss of generality. JamesBWatson (talk) 11:07, 24 December 2013 (UTC)
thread_title:  Flawed 

message: All of these above "proofs" assume that interesting and uninteresting are mutually exclusive. If we don't assume that, all sorts of numbers become both.  — Preceding unsigned comment added by 124.184.249.196 (talk) 02:12, 3 February 2014 (UTC)
thread_title:  Another attempt to resolve the paradox. 

message: The whole paradox seems to rest on the idea that being the smallest uninteresting number is an interesting quality. What if one finds the idea of the smallest uninteresting number simply tedious? 203.217.150.76 (talk) 04:26, 6 March 2014 (UTC)
thread_title:  Underlying premise 

message: "One proposed resolution of the paradox asserts that only the first uninteresting number is made interesting by that fact. For example, if 39 and 46 were the first two uninteresting numbers, then 39 would become interesting as a result, but 46 would not since it is not the first uninteresting number."
There seems to be an obvious flaw here. If 39 becomes interesting, it is no longer an uninteresting number, so it can't be the lowest uninteresting number; 46 would be the new smallest uninteresting number. 208.54.70.130 (talk) 02:07, 17 June 2015 (UTC)
thread_title:  Second-Smallest Uninteresting Number 

message: Just a quick note that, using Alex Bellos' (probably humorous) definition of "the lowest number not to have its own page on Wikipedia", the lowest uninteresting number is 1002.  And it's not just uninteresting by chance; it's uninteresting by consensus. Howicus (Did I mess up?) 01:18, 14 March 2016 (UTC)
replies: text: I was wrong, 247 is still a redirect. Howicus (Did I mess up?) 01:23, 14 March 2016 (UTC)
thread_title:  Bellos' definition 

message: I noticed that this Wikipedia page is out of date because 14972 appears as the seventh entry in a sequence so is no longer uninteresting. Searching forward from 14972, I found no uninteresting numbers until 17843. Look up http://oeis.org/search?q=17843. 17843 does appear in the OEIS, but does not appear in a integer sequence in the OEIS. It appears only as the 35th coefficient of an empirical approximation to the sequence. Does that make it interesting? Or uninteresting? Or both? Mollwollfumble (talk) 19:05, 18 November 2016 (UTC)
thread_title:  17843 is a Semi-uninteresting Number 

message: I don't see an actual method for determining this given anywhere in the article; and the definitions that are offered have no reasoning supporting them.  — Preceding unsigned comment added by 98.219.84.14 (talk) 21:59, 24 December 2020 (UTC)
thread_title:  What makes a number interesting? 

message: This page has been nominated for deletion at Wikipedia:Articles for deletion/Interesting number paradox (2nd nomination), possibly to make it the most interesting paradox without a Wikipedia article. Certes (talk) 21:25, 13 October 2022 (UTC)
thread_title:  Nominated for deletion 

message: Some sources cited in defence on the AfD page, just putting them here so they don't get lost, no idea as to the quality
  
Chaitin also calls attention to its relation to an earlier paradox of Russell on the existence of a smallest undefinable ordinal (despite the fact that all sets of ordinals have a smallest element and that "the smallest undefinable ordinal" would appear to be a definition):
 
All on JSTOR... obviously a paywall does not stop them being valid, just anyone on the outside from using them to improve the article.
82.22.50.11 (talk) 23:41, 13 October 2022 (UTC)
thread_title:  Possible Citation sources to make article better? 

message: 39 is interesting since it is the smallest number > 2 such that the Mertens function return 0 (see , I really don’t know why  David Wells thought that it is uninteresting, also see , all natural numbers up to 80 are really interesting, the next number 81 is 3^4 thus no reason to be uninteresting, but for 82 (why 82 is interesting? Companion Pell number? Number of 6-hexes? Squarefree semiprime? Ulam number? (Only mathematics-related properties are allowed, and only base-independent properties are allowed, thus you cannot say that 82 is interesting because it is the largest atomic number of a stable isotope, and you cannot say that 82 is interesting because its multiplicative inverse has only period length 5 in decimal)) 61.224.145.154 (talk) 06:31, 22 August 2023 (UTC)
thread_title:  39 is really interesting!