Editing Interesting number paradox (section)

==Paradoxical nature==
Attempting to classify all numbers this way leads to a [[paradox]] or an [[antinomy]]<ref name=":1">{{cite journal
 | last = Chaitin | first = G. J.
 | date = July 1977
 | doi = 10.1147/rd.214.0350
 | issue = 4
 | journal = IBM Journal of Research and Development
 | pages = 350–359
 | title = Algorithmic information theory
 | volume = 21}}</ref> of definition. Any hypothetical [[partition of a set|partition]] of [[natural number]]s into ''interesting'' and ''uninteresting'' sets seems to fail. Since the definition of interesting is usually a subjective, intuitive notion, it should be understood as a semi-humorous application of [[self-reference]] in order to obtain a paradox.

The paradox is alleviated if "interesting" is instead defined objectively: for example, the smallest natural number that does not appear in an entry of the [[On-Line Encyclopedia of Integer Sequences]] (OEIS) was originally found to be 11630 on 12 June 2009.<ref name="nathanieljohnston">{{cite web|url=http://www.nathanieljohnston.com/2009/06/11630-is-the-first-uninteresting-number/|title=11630 is the First Uninteresting Number|author=Johnston, N.|date=June 12, 2009|access-date=November 12, 2011}}</ref> The number fitting this definition later became 12407 from November 2009 until at least November 2011, then 13794 as of April 2012, until it appeared in sequence {{OEIS2C|id=A218631}} as of 3 November 2012. Since November 2013, that number was 14228, at least until 14 April 2014.<ref name="nathanieljohnston"/> In May 2021, the number was 20067. (This definition of uninteresting is possible only because the OEIS lists only a finite number of terms for each entry.<ref>{{Cite web |last=Bischoff |first=Manon |title=The Most Boring Number in the World Is ... |url=https://www.scientificamerican.com/article/the-most-boring-number-in-the-world-is/ |access-date=2023-03-16 |website=Scientific American |language=en}}</ref> For instance, {{OEIS2C|id=A000027}} is the sequence of ''all'' [[natural number]]s, and if continued indefinitely would contain all positive integers. As it is, the sequence is recorded in its entry only as far as 77.) Depending on the sources used for the list of interesting numbers, a variety of other numbers can be characterized as uninteresting in the same way.<ref>{{cite web|url=http://math.crg4.com/uninteresting.html|archive-url=https://web.archive.org/web/20180612143635/http://math.crg4.com/uninteresting.html|url-status=dead|archive-date=2018-06-12|title=Uninteresting Numbers|first=Charles R. |last=Greathouse IV|access-date=2011-08-28}}</ref> For instance, the mathematician and philosopher [[Alex Bellos]] suggested in 2014 that a candidate for the lowest uninteresting number would be [[224 (number)|224]] because it was, at the time, "the lowest number not to have its own page on [the English-language version of] [[Wikipedia]]".<ref>{{cite book|last=Bellos|first=Alex|others=illus. The Surreal McCoy|date=June 2014|title=The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life|edition=1st Simon & Schuster hardcover|publisher=Simon & Schuster|publication-place=N.Y.|at=pp. 238 & 319 (quoting p. 319)|isbn=978-1-4516-4009-0}}</ref> As of May 2025, this number is [[316 (number)|316]].

However, as there are many significant results in mathematics that make use of [[self-reference]] (such as [[Gödel's incompleteness theorems]]), the paradox illustrates some of the power of self-reference,{{refn|group=nb|See, for example, [[Gödel, Escher, Bach#Themes]], which itself—like this section of this article—also mentions and contains a [[wikilink]] to [[self-reference]].}} and thus touches on serious issues in many fields of study. The paradox can be related directly to Gödel's incompleteness theorems if one defines an "interesting" number as one that can be computed by a program that contains fewer bits than the number itself.<ref>{{cite book|first=Charles H. |last=Bennett |chapter=On Random and Hard-to-Describe Numbers |title=Randomness and Complexity, from Leibniz to Chaitin |editor-first=Cristian S. |editor-last=Calude |publisher=World Scientific |year=2007 |doi=10.1142/9789812770837_0001 |pages=3–12 |isbn=978-9-812-77082-0 |oclc=173808093 |author-link=Charles H. Bennett (physicist) |editor-link=Cristian S. Calude }} Originally circulated as a preprint in 1979.</ref> Similarly, instead of trying to quantify the subjective feeling of interestingness, one can consider the length of a phrase needed to specify a number. For example, the phrase "the least number not expressible in fewer than eleven words" sounds like it should identify a unique number, but the phrase itself contains only ten words, and so the number identified by the phrase would have an expression in fewer than eleven words after all. This is known as the [[Berry paradox]].<ref>{{Cite book |last=Yanofsky |first=Noson S. |url=https://www.worldcat.org/oclc/857467673 |title=The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us |date=2013 |publisher=[[MIT Press]] |isbn=978-1-4619-3955-9 |location=Cambridge, Massachusetts |pages=26–28 |oclc=857467673}}</ref>