Story Size Abstract: The shortest stories are zero words long. T here is no maximum length. Take a strip of paper with 'once upon a time there' written on one side and 'was a story that began' on the other. Twisting the paper and fastening the ends produces John Barth's Frame-Tale, which prefixes a token of 'once upon a time there was a story that began' to itself.1 According to its author, Frame-Tale is "... the shortest story in the English language (ten words); on the other hand, it's endless."2 This paper argues Frame-Tale is neither the shortest nor the longest story: zero is the minimum story length, and there is no maximum. One clarification. This paper contributes nothing original, even pedagogically, to mathematics or its philosophy, and its contributions to the philosophy of literature are merely marginal. The purpose of the paper is not to use examples from literature to cast light on philosophical problems. On the contrary, the purpose of the paper is the opposite: to use mathematical and philosophical ideas for the creation of literature. Frame-Tale is an example of the fascinating fiction which may be created in this way, but, I shall argue, it falls far short of exhausting the possibilities in the area. The application of mathematics to the elaboration of novel literary forms has been extensively explored by the Oulipo, a collective writing mainly in French, but to my knowledge they've overlooked the possibility of exploiting transfinite numbers.3 The infinite as a theme in literature has been well explored by Jorge Luis Borges. I've taken his advice when he says "It is a laborious madness and an impoverishing one, the madness of composing vast books – setting out in five hundred pages an idea that can be perfectly related orally in five minutes. The better way to go about it is to pretend those books already exist, and offer a summary, a commentary, on them."4 There are many microfictions shorter than ten words. Augusto Monterroso's story The Dinosaur, for example, consists of the eight words: 'When he awoke, the dinosaur was still there.'5 Ernest Hemingway is purported to have written a story of just six words: 'Baby's shoes: brand new, never worn.'6 And Forrest Ackerman's story Cosmic Report Card: Earth consists of a single letter: 'F'.7 According to its author Cosmic Report Card: Earth is "... what must be the world's shortest science fiction story – one letter of the alphabet for which I got paid a hundred dollars."8 2 But the shortest stories are less than one letter long.9 Metamicrofiction, for example, is a short story which consists of no letters and exactly zero words.10 Just as Frame-Tale is an infinitely long story about an infinitely long story, Metamicrofiction is a zero word story about a zero word story.11 Since Metamicrofiction consists of no words, letters or other symbols at all, no story is shorter. (Metamicrofiction does have one word in its title. But one can imagine the publication of an untitled sequel, with a similar theme.12) So zero is the minimum story length. It might be objected that Metamicrofiction and its sequel are not stories, since they lack, for example, traditional plot and characterization. But Frame-Tale also lacks traditional plot and characterization. And insofar as Frame-Tale is a traditional narrative whose protagonist is a story which heroically persists despite the odds against it (it isn't), Metamicrofiction is a traditional narrative whose protagonist is a story which tragically ceases despite the odds in its favour (it isn't either). So insofar as Frame-Tale is a story, Metamicrofiction is a story as well. Some definitions of story exclude Frame-Tale and Metamicrofiction simply because of their lengths. Aristotle's, for example, defines a story as the imitation of a whole and complete action, which has a beginning, middle, and ending, as well as an appropriate length.13 Frame-Tale is excluded because it has no beginning nor end and is inappropriately long, whereas Metamicrofiction is excluded since, as well as having no beginning nor end it has no middle, it is inappropriately short. But this tells more against Aristotle's definition than it does against the stories. Other definitions of story exclude Frame-Tale and Metamicrofiction because they are metafictions, and not because they are microfictions. Most definitions require a causal connection between the events of the story, which excludes Frame-Tale and Metamicrofiction because they don't feature causation in any obvious way.14 This is an important difference between Frame-Tale, Metamicrofiction and other stories. But whether it warrants withholding the word 'stories' is a verbal question that there is no need to resolve.15 There are many novels longer than ten words. Marcel Proust's A la recherche du temps perdu, which consists of approximately nine million six hundred and nine thousand characters including spaces, is often estimated to be the longest.16 Someone reading Frame-Tale at the same time per character as someone reading A la recherche du temps perdu would complete about two hundred and eight thousand eight hundred 3 and ninety-one rotations in the time taken to complete A la recherche du temps perdu. Nevertheless, the reader of Frame-Tale will still have more to read after the reader of A la recherche du temps perdu has finished. Some poetry is much longer than A la recherche du temps perdu. Raymond Queneau's A Hundred Thousand Billion Poems, for example, consists of ten fourteen line poems with identical rhyme schemes.17 Cutting between the lines allows them to be recombined into a total of ten to the power of fourteen or one hundred thousand billion poems. Queneau estimated that "... someone reading the book 24 hours a day would need 190,258,751 years to finish it", implying a reading speed of one poem per minute, except for rest on leap days.18 Nevertheless, even if someone finished reading A Hundred Thousand Billion Poems at this speed, someone reading Frame-Tale at the same speed would still have more to read before and after. Michèle Métail is composing a poem called Compléments de noms, inspired by the long German word 'donaudampfschifffahrtsgesellschaftskäpitan'.19 The first few lines run: le capitaine de la compagnie des voyages en bateau à vapeur du danube la femme du capitaine de la compagnie des voyages en bateau à vapeur la fille de la femme du capitaine de la compagnie des voyages en bateau ... and so on, where each line is succeeded by one dropping the last possessive phrase and prefixing a new one. The poem is reputed to be more than twenty thousand lines long, but is intended to be infinite.20 Whereas Queneau's instructions suffice to determine which poems are part of A Hundred Thousand Billion Poems, Métail's instructions underdetermine which line of Compléments de noms is next, so Métail's intention for Compléments de noms to be infinite is unrealised, and the poem is incomplete. (It might also be worried that Compléments de noms cannot be infinite, since each line requires new vocabulary, but the vocabulary of French is finite. However, Métail sidesteps this worry by adopting vocabulary from other languages, and one may expect the growth of vocabulary in all languages to outstrip the growth of the poem.21) Nevertheless, there are novels and poems which are infinitely long. Take, for example, ω, which is a verse novel consisting of the following sentences: I write I write I write 4 I write I write I write ... and so on, where each successive line prefixes 'I write' to its predecessor.22 These instructions, like Queneau's for A Hundred Thousand Billion Poems, suffice to determine the next line of ω. So like Frame-Tale, ω is endless. Nevertheless, ω and Frame-Tale are not comparable with respect to length. Since ω and Frame-Tale are both endless, someone who reads a line of ω in the same time as someone who reads one rotation of Frame-Tale will continue reading for as long. But since Frame-Tale is beginningless, whereas ω is not, someone who reads one line of ω in the same time as someone who reads one rotation of Frame-Tale won't have been reading for as long. The reader of Frame-Tale will already have been reading from eternity when the reader of ω has just begun. In other words, if ω and Frame-Tale were comparable with respect to length, then it would be possible for us to compare their lengths by beginning reading at the same time and at the same speed, and discovering whether one of us reads for longer than the other. But although neither of us would read for longer than the other – since neither of us would ever stop reading if we maintained constant speed – we cannot begin reading at the same time, because Frame-Tale has no beginning to begin from – to read it at constant speed, one has to have always been already reading it. Reordering ω so that the lines in which 'I' occurs an even number of times appear first, in descending order according to the number of times which 'I' appears, and the lines in which 'I' occurs an odd number of times appear second, in ascending order according to the number of times which 'I' appears, produces ω* + ω, a novel which is equal in length to Frame-Tale.23 Like Frame-Tale, ω* + ω has no beginning, since before every line there is either a shorter line in which 'I' occurs an odd number of times or a longer line in which 'I' occurs an even number of times.24 Frame-Tale and ω* + ω are equal and comparable in length, since someone who reads a line of ω* + ω in the same time as someone who reads one rotation of Frame-Tale will continue reading for as long, since ω* + ω and Frame-Tale are both endless, but will also have been reading for as long, since ω* + ω and Frame-Tale are also both beginningless. In other words, if you and I read ω* + ω and Frame-Tale at the same constant speed, then we will both continue reading for ever, but we will both also have always been already reading it. Taking 'I write I write' from ω* + ω and placing it at the end produces ω* + ω + 1, a novel which is one sentence longer than ω* + ω, and so longer than Frame5 Tale. ω* + ω + 1 is longer than ω* + ω because every sentence in which 'I' occurs an odd number of times in ω* + ω corresponds to the same sentence in ω* + ω + 1, and every sentence of ω* + ω in which 'I' occurs an even number of times corresponds to a sentence in which 'I' occurs twice more in ω* + ω + 1, but no sentence in ω* + ω corresponds to the final sentence of ω* + ω + 1. In other words, ω* + ω + 1 is longer than ω* + ω since if you and I read and have always been reading ω* + ω + 1 and ω* + ω at the same constant speed, so that every time I read a line of ω* + ω you read a line of ω* + ω + 1, then although I will always have more lines to read, every line of ω* + ω will eventually be read by me, whereas the last line of ω* + ω + 1, 'I write I write', will never be read by you. Since it would take forever to read the lines of ω* + ω + 1 which precede 'I write I write', there would be no time left to read the last line. So Frame-Tale is not the longest story. It might be objected that ω, ω* + ω, and ω* + ω + 1 can't be read and written in full. But similar objections can be raised about Frame-Tale and A Hundred Thousand Billion Poems: Frame-Tale can be written but not read in full and the one hundred thousand billion poems in A Hundred Thousand Billion Poems all have some chance of being read, but most never will be. Likewise, flipping a coin until it lands tails and then writing and reading 'I write' as many times as it landed heads gives every sentence in ω, ω* + ω, and ω* + ω + 1 a chance of being read. It doesn't matter that most of them never will be.25 It might also be objected that whereas it's possible in principle to read A Hundred Thousand Billion Poems in a finite but long period of time, it's impossible even in principle to read Frame-Tale, ω, ω* + ω, or ω* + ω + 1 in a finite period of time. But it is possible in principle to read ω in a finite period of time: if one reads the first sentence in half a minute, the second sentence in a quarter of a minute, the third sentence in an eighth of a minute, ... and so on, then one will be finished before a minute. Of course, it's not possible to read, say, the sixth sentence in less than a second but, like finishing A Hundred Thousand Billion Poems before death, this is a mere impracticability. Likewise, it is possible in principle to read Frame-Tale in a finite period of time: if one reads two rotations in a quarter of a minute each, before reading one in an eighth of a minute as well as after reading one in an eighth of a minute, before reading 6 one in a sixteenth of a minute as well as after reading one in a sixteenth of a minute, ... and so on, then one will be finished reading before a minute. It may be thought one can't have finished Frame-Tale in less than a minute, since one must have broken off after the last rotation. But there was no last rotation, since after every rotation in the last half minute, there was another which was half as long. Similarly, one needn't have begun reading in medias res, because there was no first rotation either: before every rotation in the first half minute, there was another which was half as long.26 It's possible in principle to read ω* + ω in the same finite period of time: if one reads two rotations in a quarter of a minute each, before reading one in an eighth of a minute as well as after reading one in an eighth of a minute, ... and so on, then one will be finished reading before a minute. And if ω* + ω + 1 is read at the same speed, it will be finished in almost the same finite amount of time, with only the addition of the time taken to read the final sentence. (Some of the novels mentioned below may have more elements than there are times, so whether it's possible to read one of these novels depends on whether it's possible for time to have more moments than the novel has elements.) Let us say that a novel is well-ordered if and only if all of its abridgements have a first part (and let us say that an abridgement of a novel is any collection of the parts of the novel in their usual order, including the whole novel, but excluding the collection of none of its parts).27 ω, for example, is a well-ordered novel, since every abridgement of ω has a first part: its shortest sentence. But ω* + ω, on the other hand, isn't a wellordered novel, since ω* + ω is an abridgement of itself with no first sentence. FrameTale, likewise, is not well-ordered, because it has no beginning.28 Every two well-ordered novels are comparable in length, because the first part of the first novel corresponds to the first part of the second novel, and the first part of the rest of the parts (if any) of the first novel corresponds to the first part of the rest of the parts (if any) of the second novel, and so on. If every part of the first corresponds in this way to a part of the second and vice versa, then the novels are the same length, whereas if some part of the first corresponds to no part of the second, then the first is longer, and vice versa.29 Taking 'I write I write' from ω and placing it at the end, for example, produces ω + 1, a novel which is one sentence longer than ω. ω + 1 is one sentence longer than ω, because each sentence of ω corresponds to a sentence of ω + 1, but the 7 last sentence of ω + 1 does not correspond to any sentence of ω: 'I write' in ω corresponds to 'I write' in ω + 1, 'I write I write' in ω corresponds to 'I write I write I write' in ω + 1, 'I write I write I write' in ω corresponds to 'I write I write I write I write' in ω + 1, ... and so on, but 'I write I write' in ω + 1 does not correspond to any sentence in ω. Taking the second and fourth sentences from ω and adding them to the end produces ω + 2, a novel two sentences longer than ω. Taking the second, fourth and sixth sentences produces ω + 3, a novel three sentences longer, taking the second, fourth, sixth and eighth produces ω + 4, ... and so on. Taking every second sentence from ω and adding them to the end produces ω + ω or ω.2, which is twice as long as ω. But even ω.2 is not the longest novel: taking it's second sentence and adding it to the end produces ω.2 + 1, a novel one sentence longer still.30 In general, for any novel in this series, its sequel is produced, if it has no last sentence, by taking the second sentence from the beginning and adding it to the end or, if it has a last sentence, by counting backwards to find the number of sentences before one with no immediate predecessor, and then counting forwards by twice that number to find the sentence to take to the end. Counting back from the end of ω + 4, for example, one finds four sentences before one with no immediate predecessor, so one takes the eighth sentence from ω + 4 to the end to produce ω + 5.31 The novel ω.2, although it is longer than ω, ω + 1, ω + 2, ..., and so on, is not the sequel of any of them; it is produced by collating all its prequels. In general, for an endless sequence of sequels of ω, in which all prequels of all novels in that sequence are included, the next novel is produced by first taking the sentences not preceded by a sentence with no immediate predecessor except the first in every novel, and then all sentences which are preceded by a sentence with no immediate predecessor except the first in any novel, as well as the second sentence with no immediate predecessor.32 The next novel after the endless series ω, ω + 1, ω + 2, ..., for example, is the novel consisting of all the sentences with an odd number of occurrences of 'I', since these occur in all of ω, ω + 1, ω + 2, ... without being preceded by a sentence with no immediate predecessor except the first, followed by all the sentences with an even number of occurrences of 'I', since all of these except 'I write I write' are preceded by 'I write I write' in ω, ω + 1, ω + 2, ..., and since 'I write I write' has no immediate predecessor (and is not the first sentence with no immediate predecessor). 8 These two processes produce the following series of longer and longer novels, of which ω is merely the shortest: ω, ω + 1, ω + 2, ..., ω.2, ω.2 + 1, ..., ω.3, ..., ω2, ..., ω3, ..., ωω, ... Though the sentences of ω may be stretched into novels of greater and greater length, there is a sense in which all these novels, as well as ω* + ω and ω* + ω + 1, all have the same size, because they were all produced by reordering the sentences in ω, and so all contain the same number of sentences.33 Taking each novel in this series as the chapters of a larger novel produces ω1, which encompasses all of them. ω1 is longer than any chapter of ω1, because if the first sentence of the chapter corresponds to the first chapter of ω1, then the succeeding sentences of the chapter correspond to succeeding chapters of ω1, but a succeeding chapter of ω1 does not correspond to any sentence. Every sentence in chapter ω, for example, corresponds to a chapter preceding ω.2, since 'I write' corresponds to ω, 'I write I write' to ω + 1, 'I write I write I write' to ω + 1, ... and so on, but no sentence of ω corresponds to chapter ω.2 itself. Not only is ω1 longer than any of its chapters, it is also larger than any of its chapters, since ω1 has more chapters than any of its chapters has sentences. For suppose ω1 had the same number of chapters as its chapters have sentences. Then it would be possible to reorder the sentences of ω so that each sentence corresponds to a chapter of ω1. But if it were possible to reorder ω so that each sentence corresponded to a chapter of ω1, then since ω1 encompasses as its chapters all novels stretched from the sentences of ω, this reordering would be a chapter of ω1 which is also the length of ω1. But we have already showed that ω1 is longer than any of its chapters.34 Taking the second chapter from ω1 and adding it to the end produces ω1 + 1, taking every second chapter of the chapters from ω1 preceding ω.2 produces ω1 + ω, ... and so on. In general, applying the two procedures mentioned above to chapters instead of sentences produces the following series of longer and longer novels: ω1, ω1 + 1, ..., ω1 + ω, ..., ω1.2, ..., ω12, ..., ω1ω, ..., ω1ω1, ... Though the chapters of ω1 may be stretched into novels of greater and greater lengths, all these novels are the same size, because they are all composed of the same number of chapters, which are all composed of the same number of sentences. Taking all the preceding novels as the volumes of a larger novel produces ω2, which encompasses all of them. ω2 is longer than any volume of ω1, because if the first chapter of that volume corresponds to the first volume of ω2, then the succeeding 9 chapters of that volume correspond to succeeding volumes of ω2, but a succeeding volume of ω2 does not correspond to any chapter. Every chapter in volume ω1, for example, corresponds to a volume preceding ω1, since ω corresponds to ω, ω + 1 to ω + 1, ... and so on until ω.2, which corresponds to ω.2, ..., and so on, but no chapter of ω1 corresponds to volume ω1 itself. ω2 is not only longer than any of its volumes, but also larger than any of its volumes, since ω2 has more volumes than any of its volumes has chapters, for the same reason ω1 has more chapters than any of its chapters have sentences. Reordering the volumes of ω2 produces longer novels of the same size, which with preceding novels can be gathered into ω3, which is also longer and larger than any of them. And reordering ω3 produces longer novels, which with preceding novels can be gathered into ω4, ..., leading to ever larger novels ω5, ω6, ... ωω, ωω+1, ..., ωω.2, ..., and so on. In general, by applying the first procedure mentioned above to the direct parts of a novel one produces a sequel to that novel which is one part longer. By applying the second procedure mentioned above to the direct parts of an endless sequence of sequels of those novels, which includes all prequels of all novels in that sequence, one produces the next novel in the series. And by concatenating together all the novels in one of these series into a single novel one produces the first novel of the next series, which is not only longer but also larger than all of them. It might be thought that it's possible to collate all novels composed in this way into a single novel which encompasses them all, and so is longer than all the novels. But this is not possible. For suppose it were possible to gather all novels composed in this way into a single novel which encompasses them all, and so is longer than all of them. Then by taking this novel's second element and adding it to the end, it would be possible to compose an even longer novel. But since this novel was composed in the same way, this would contradict the supposition we had gathered all novels composed in this way into a single novel which is longer than them all.35 So there is no longest story. In every novel considered so far, the elements possess a specific order, and reordering the elements produces a different novel. But for some novels, the elements possess no specific order, and reordering the elements simply produces a different version of the same novel. Bryan Johnson's The Unfortunates, for example, is published in a box, so the unbound sections, except the first and last, can be read in any order.36 Likewise, 10 Marc Saporta's Composition No. 1 consists of a box of unbound pages, which should be shuffled and read in random order.37 Abstracting from the order of sentences in ω produces 0א, a novel in which the sentences can be read in any order.38 has no determinate length, but since it has the same number of sentences 0א as ω, it has the same size as ω, and as every other novel produced by reordering the sentences of ω. Abstracting from the order of chapters in ω1 produces 1א, which has the same size as ω1 and every other novel produced by reordering the chapters of ω1. Abstracting from the order of volumes in ω2 produces 2א, which has the same size as ω2 and every other novel produced by reordering the volumes of ω2. ... And so on.39 But abstracting from the order of the elements in the novels discussed so far is not the only way to produce novels larger than 0א. Taking all subsets of sentences in in no particular order, as the paragraphs, in no particular order, of a new novel ,0א produces 20א2 .0א is larger than 0א, because 20א has more paragraphs than 0א has sentences. For suppose the number of paragraphs in 20א were the same as the number of sentences in 0א. Then it would be possible to head each paragraph of 20א with a sentence of 0א. But this is not possible. For suppose every paragraph in 20א were headed with a sentence of 0א. Then take the paragraph consisting of all and only the sentences of 0א not contained in the paragraphs they head. This paragraph could not be headed by a sentence it contains, since it contains only sentences heading paragraphs not containing them. Nor could it be headed by a sentence it does not contain, since it contains all sentences heading paragraphs not containing them. So this paragraph could not be headed by any sentence, which contradicts our supposition.40 Since 20א is larger than 0א, and since 0א is the same size as ω, 20א is larger than ω. So if we can choose a chapter of 20א to be the first, and then choose another chapter to be next, and then the next, ... and then choose a chapter to be next after all these chapters, ... and so on until the chapters of 20א are well-ordered, then 20א will be longer than ω and so at least as long as ω1.41 Since ω1 is the same size as 1א, it would follow that 20א is also at least as large as ω1 and 1א. But whether 20א is exactly the same size as depends on the continuum hypothesis, a famously unresolved question.42 1א Regardless of whether 20א is comparable in size with 1א or any larger novel mentioned so far, 20א cannot be the largest novel, since a larger novel is produced by 11 taking every subset of the paragraphs in 20א, in no particular order, as the chapters, in no particular order, of a new novel. This novel has more chapters than 20א has paragraphs, since if its chapters numbered the same or less than the number of paragraphs in 20א, it would be possible to head all of them with a paragraph from 20א. But this is not possible. For suppose all its chapters were headed with a paragraph of 20א. Then take the chapter consisting of all and only the paragraphs of 20א which aren't contained in the chapters they head. This chapter couldn't be headed by a paragraph it contains, since it contains only paragraphs heading chapters which don't contain them. But nor could it be headed by a paragraph it does not contain, since it contains all paragraphs heading chapters which don't contain them. So this chapter could not be headed by any paragraph.43 It might be thought that there's a single novel which has every novel as one of its chapters, and so is the largest novel. But there cannot be such a novel. For consider the novel which has as its volumes every subset of these chapters. This second novel must have more volumes than the first has chapters, since if its volumes numbered the same or less than chapters in the first, it would be possible to preface every volume of the second with a chapter of the first. But this isn't possible. For suppose all its volumes were prefaced by a chapter. Then take the volume consisting of all and only the chapters not contained in the volumes they preface. This volume couldn't be headed by a chapter it contains, since it contains only chapters prefacing volumes which don't contain them. But nor could it be prefaced by a chapter it does not contain, since it contains all chapters prefacing volumes which don't contain them. So this volume could not be prefaced by any chapter.44 So there is no largest story.45 1 John Barth, Lost in the Funhouse (New York: Doubleday, 1968), pp. 1-2. As well as the issues discussed in this paper, Frame-Tale raises some interesting questions in the philosophy of language. See my paper "A Never-Ending Story" in Croatian Journal of Philosophy 14(40) (2014), pp. 111-120 for some discussion of these issues. 2 Lost in the Funhouse, p. vii. 3 For accounts of the Oulipo's works see especially Warren Motte (ed.), Oulipo: A Primer of Potential Literature (Lincoln: University of Nebraska Press, 1986), Oulipo 12 Laboratory (London: Atlas Press, 1995), Harry Mathews and Alistair Brotchie (eds.), Oulipo Compendium (London: Atlas Press, 1998), and "The State of Constraint: New Work by Oulipo", McSweeney's 22 (2006). 4 Jorge Luis Borges, "The Garden of Forking Paths" in Collected Fictions, trans. Andrew Hurley (London: Allen Lane, 1999), p. 67. 5 Augusto Montessori, Complete Works and Other Stories, trans. Edith Grossman (Austin: University of Texas Press, 1996), p. 42. 6 Wired Magazine 14 (November 2006). Amihud Gilead discusses these examples in "How Few Words Can a Short Story Have?" Philosophy and Literature 32 (2008), pp. 119-129. The attribution of the story to Hemingway is dubious: see Frederick A. Wright, "The Short Story Just Got Shorter: Hemingway, Narrative, and the Six-Word Urban Legend", Journal of Popular Culture 47(2) (2014), pp. 327-340. 7 Forrest Ackerman, "Cosmic Report Card: Earth", Vertex Magazine 1 (January 1973). See also "Going for the Limit" in Oulipo Compendium, pp. 177-178, which mentions "A poem comprising of a single letter: T" amongst twenty-five other possibilities. 8 Forest Ackerman, "Through Time and Space with Forry Ackerman", Mimosa 16 (1994), pp. 4-6. 9 Zero word poems are discussed briefly in Oulipo Compendium, p. 251. Paul Fournel's "Suburbia", trans. Harry Mathews in Oulipo Laboratory, pp. 37-60 purports to be a zero word novel. Craig Dworkin, No Medium, (Cambridge, Mass: MIT Press, 2013) discusses similar examples in several media. And see also Dennis Upper, "The Unsuccessful Self-Treatment of a Case of "Writer's Block"", Journal of Applied Behavior Analysis 7(3) (1974), p. 497. For discussion of related examples in music, see Andrew Kania, "Silent Music", Journal of Aesthetics and Art Criticism 68(4) (2010), pp. 343-353. 10 Journal of Microliterature (July 2012). It might be objected that Metamicrofiction is not about itself (and so mistitled) because it has no words. But so long as it exists, it is capable of being about something, and so capable of being about itself. 11 This interpretation of Metamicrofiction is supported by its title, according to which it is about a short fiction – since it is about a short fiction, and it is a short fiction, it is reasonable to conclude that it is the short fiction it is about. 13 12 Jorge Luis Borges and Adolfo Bioy Casares describe the publication of an untitled zero word story in "A List and Analysis of the Sundry Books of F. J. C. Loomis", Chronicles of Bustos Domecq, trans. Norman Thomas Di Giovanni (New York: Dutton, 1979), pp. 49-55. See also "", Philosofict 1 (March, 2013). 13 Aristotle, Selections, trans. Terence Irwin and Gail Fine (Indianapolis: Hackett, 1995), pp. 544-547. Aristotle's uses 'muthos' for story or plot. See Selections, p. 544. 14 See, for example, Noël Carroll, "The Narrative Connection" in Beyond Aesthetics (Cambridge: Cambridge University Press, 2001) and Gregory Currie, Narratives and Narrators (Oxford: Oxford University Press, 2010). Carroll distinguishes a narrative, which involves causation, from a story, which is merely a sequence of events. It's not obvious to me whether Frame-Tale or Metamicrofiction are stories even in this sense. 15 Kendall Walton distinguishes between the standard features of a category – those features which are necessary for a work to have in order to belong to the category – and the variable features of a category – those features which are neither necessary for a work to have to belong to the category nor sufficient for the a work not to belong to the category. Walton admits that which features of a category are standard is vague. In the case of stories, it is vague whether it is a standard feature of a story that it has a beginning middle and end, or is of an appropriate magnitude, or has traditional plot and characterization, so Frame-Tale and Metamicrofiction are borderline cases of the category of stories. See Kendall Walton, "Categories of Art", Philosophical Review 79(3) 1970, p. 339. 16 Marcel Proust, A la recherche du temps perdu (Paris: Bernard Grasset, 1913-27). The character count is given in Guinness World Records (London: Guinness World Records, 2011), p. 163. 17 Raymond Queneau, "A Hundred Thousand Billion Poems", trans. Stanley Chapman in Oulipo Compendium, pp. 15-33. 18 Oulipo Compendium, p. 14. 19 Michèle Métail, "L'infini moins quarante annuities", Le cahier du refuge 214 (September 2012) and Michèle Métail, 2888 Donauverse aus einem unendlichen Gedicht (Vienna: Edition Korrespondenzen, 2006). Similar examples are discussed by Izzy Grinspan in "Verse by the Yard", Poetry Foundation (April, 2008). 20 Daniel Levin Becker, Many Subtle Channels: In Praise of Potential Literature (Harvard: Harvard University Press, 2012), p. 200. Métail writes "Il me dit « c'est très 14 bien, mais il n'y en a pas assez ». Qu'à cela ne tienne, le poème sera donc infini!", "L'infini moins quarante annuities", p. 5. 21 "L'infini moins quarante annuities", p. 6. 22 The opening lines of ω resemble the opening lines of Georges Perec's Species of Spaces, trans. John Sturrock (London: Penguin, 1997). Although the title resembles that of Perec's W, or the memory of childhood, trans. David Bellos (London: Harvill, 1988), ω is named after the ordinal number which it's like. An accessible introduction to ordinal numbers is in Bertrand Russell's Introduction to Mathematical Philosophy (London: George Allen and Unwin, 1919), pp. 89-96. 23 ω* + ω is named for its likeness to the series of negative and positive integers. See Bertrand Russell, Principles of Mathematics (Cambridge: Cambridge University Press, 1903), p. 320. 24 Roy Sorenson writes "...there are palindromes of every finite length. Therefore, there are infinitely many palindromes. Yet no palindrome is infinitely long. After all, a palindrome must read the same way backwards. If the sentence went on forever, where would one begin the reversal?" ("A Plenum of Palindromes for Lewis Carroll", Mind 109 (2000), p. 20). But ω* + ω illustrates that if it had no beginning as well as no end, a palindrome could be infinitely long. 25 By stipulating that the first half of a blank line abbreviates 'I write', the first half of the second half abbreviates 'I write I write', the first half of the third half abbreviates 'I write I write I write'... and so on, ω can also be abbreviated into a single line. See Roy Sorenson, "Blanks: Signs of Omission", American Philosophical Quarterly 36(4) (1999), p. 310. ω* + ω and ω* + ω + 1 can be abbreviated via similar stipulations. If space is continuous, then some stories of size 20א, but no greater, may also be abbreviated in similar ways. And whether stories of size larger than 20א can be abbreviated in this way in principle depends on whether there can in principle be more than 20א spaces in which to abbreviate them. 26 Reading ω, Frame-Tale, ω* + ω and ω* + ω + 1 in this fashion involves the performance of a super task. See James Thomson, "Tasks and Supertasks", Analysis 15(1) (1954), pp. 1-13, which argues for the in principle impossibility of these tasks, and Paul Benacerraf, "Tasks, Super-Tasks, and the Modern Eleatics", Journal of Philosophy 59(24) (1962) pp. 765-784, which argues – correctly I think – for their in principle possibility. 15 27 Introduction to Mathematical Philosophy, pp. 92-93. 28 Frame-Tale's order is cyclical, see Introduction to Mathematical Philosophy, pp. 40-41. 29 For detailed explanation see Karel Hrbacek and Thomas Jech, Introduction to Set Theory, 3rd Edition, (New York: Marcel Dekker, 1999), pp. 104-106. 30 Introduction to Mathematical Philosophy, pp. 90-92. 31 This is a variation of Cantor's first principle of generation. See Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans. Philip Jourdain (London: Open Court, 1915), p. 169. 32 This is a variation of Cantor's second principle of generation. See Contributions to the Founding of the Theory of Transfinite Numbers, p. 169. 33 Introduction to Mathematical Philosophy, p. 92. 34 Introduction to Set Theory, pp. 103-110 and pp. 129-132. 35 This is a manifestation of the Burali-Forti Paradox. See Principles of Mathematics, p. 324 and Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, Foundations of Set Theory, 2nd Edition (Amsterdam: Elsevier 1973), p. 8. 36 Bryan Stanley Johnson, The Unfortunates (London: Secker & Warburg, 1969). 37 Marc Saporta, Composition No. 1, trans Richard Howard (London, Visual Editions, 2011). 38 Though the title resembles that of Borges' "The Aleph" in Collected Fictions, pp. is named for the cardinal number to which it's similar. See Introduction to 0א ,274-85 Mathematical Philosophy, p. 83. 39 Introduction to Mathematical Philosophy, p. 92. 40 This is an application of Cantor's diagonalisation argument. See Introduction to Mathematical Philosophy, pp. 85-86. 41 This argument depends on the axiom of choice. See Introduction to Mathematical Philosophy p. 92, pp. 122-124. 42 See Introduction to Set Theory, p. 101. 43 A possible world is sometimes defined as a complete and consistent novel – a novel in which all the sentences are compatible with each other, but are jointly incompatible with any sentence not in the novel. A common criticism of this definition is that since in an ordinary language there are only 0א sentences, it follows that there will be at most 20א possible worlds (since that is the number of sets of the 0א sentences). This 16 objection is not undermined by the point that there is a novel with more chapters than there are paragraphs in 20א, since the number of these chapters exceeds the number of complete and consistent novels only by containing many repetitions of the same 0א sentences, and so the different chapters do not correspond to distinct possibilities. See Richard Jeffrey, The Logic of Decision (Chicago: University of Chicago Press, 1965), p. 208 and David Lewis, Counterfactuals (Harvard: Harvard University Press), p. 90 and On The Plurality of Worlds (Oxford: Blackwell, 1986), pp. 143-144 for the definition and criticism respectively. For a surprising connection between the philosophy of possible worlds and the Oulipo, see Jacques Roubaud, On The Plurality of Worlds of Lewis, trans. Rosmarie Waldrop (Normal: Dalkey Archive Press, 1995). 44 This is a manifestation of Cantor's Paradox. See Principles of Mathematics, pp. 362-368. 45 I would like to thank David Bourget, David Chalmers, Alex Sandgren and WengHong Tang, as well as Sam Baron, Daniel Levin Becker, Derek Baker, Mark D'Cruz, John Paul Foenander, Daniel Friedrich, Axel Gelfert, John Holbo, Cliff Kerr, June Mahadevan, Daniel Nolan, Shaun Oon, Angela Mendelovici, Michael Pelczar, Andrew Hao-Yen Shih, Neil Sinhababu, Sarindranath Tagore, Liling Tan, David Wall, Zach Weber, Cecilia Wee, Brenton Welford, Zi Wei Yap, anonymous referees for Philosophical Papers and audiences at the Australasian Association of Philosophy conference at the University of Victoria Wellington, the University of Sydney, and Humboldt University Berlin for help with this paper.