Answering Hilberts 1st Problem

Answ ering Hil bert’s 1st Problem Charles Sauerb ier (19 March 202 1) Dogma gives a charter to mistake, but the very b reath of science is a contest wit h mistake, and must keep the conscience alive. ~ George Eliot Abstract Hilbert’s first problem is of impo rtance in relation to work being done i n computational systems. It is the que stion of equipollence of natural and real numbers. By constr uc tion equipollence is established for real numbers in open interval (0, 1 ) and natural numbers and, from such to all real numbers. Construction stands in contradiction of the generally accepted diagonal argument o f Cantor. Mathem atics being irrefutable, in absence r ejection of all theory of mathematics and logic, th e problem exists in acceptan ce ; that itself arises of mo re fundamental a problem in science genera lly. The problem within Hilbert’s prob l em is of Schopenhauer’ s, et al, “will and representation” born. Keywords Number Theory, Set The ory, Card inality, Enumeration Algorithms, Recursive Enumerati on Background Hilbert in l ecture, as recorded in [1], prod uced at the turn of the last century a list of probl ems in mathematics. The first problem in his list is titled “ Cantor’s P ro blem of th e Cardina l Number of the Continuum ”. By al l avai lable references this remains an open probl em a century after being po sited. The question is itself cent ral to many open problem s in computational systems theory. In answering Hilbert’s challenge so distant in time o ne ope ns new problems, an d upends some of what has come to be accepted. The question of whether th e set of real num bers is denum e rable pr edates both Cantor and Hilbert. Smorynski in [2] pr esents Cantor ’s “ Pairing Fun ction ”, that is an element in proof of work presente d here. [3] is a general reference for autho r’s substantive recollection o f concepts that is of parti cular note for breadth of content of rel evance to presented work. Problem within the Problem “A given opin ion, as held by se veral individuals, even when of t he most congenial views, is as distinct as ar e their face s.” ~ St. John Henry Cardina l Newman Such is at the root of the argument in proof of what is written. Homage here need be pa id to Hofstadter’s classic [4]; in which pages one finds per haps the most eloquent exposition of what here is onl y briefly touched, a fter Loui s Carol, in the vei n of the insightfu l comment of Cardin a l Newman and others. It is hopefully with suffic i ency nec essary to mo ve beyond the inherent fallacy of re asoning givin g rise to i t. A Gedankenexperime nt if you will: Yo u come upon a literal “.” point in the em ptiness of space. From it, as far int o the distance as distant can be you obs erve this tiny machine tirelessly and endlessly extending a string of what by all appearanc e is the digit “3” on your left from the point; only to observe a n identical tiny machi ne is sim ilarly do ing a t the end o f identical stri ng extending to your right. Someone then appears, of n owhere parti cular, to inquire of what purpose th ese tiny machines, and of what is t heir intent. Y our first th oug ht of wh at those two machines are doing? Therein rests foundati on of Hilbert’ s consternation. Our two tiny machines are in fact do ing nothing more than what is literally observed: extending from a si ngle point in space two strings of what in our ideati on repres ent a digit – n amely “3”. All else of any respo nse arises of an individual’s menta l notions and processes – of opinion s held , as i t be. In [5] Arth ur Schopenh auer comments at first line what is, irrespective opi nions held, reality: “The world is my r epresentatio n” 1 . He gives in that work worthy discussion of the m etap hysical problem within Hilbert’s 1st problem. While in [6] an d in [7] o ne is int roduced to ho w sc i ence goes so awry that Carrol and Hofstadter should have s ufficient material of wh ich to write such beguiling text, so entertaining of the mind. If you were tol d our two tiny machines, having h eard Einstei n’s conjectur e the universe was curved in mann er of a giant s phere, were intent on encircling it with means of conv eyance for which the “rail ” was composed of what did to all observ ers appear to be the digit “3”? What then m ight your will repre sent of your expe rience? Thus only t o have a second figure appear, of nowhere parti cul ar, to explain the t wo tiny mac hines are writing the results in bo th directions from t heir origin of ∑ 3 × 10    , inv iting you to draw nearer th at point in spa ce; where upon you become aw are t hat point is an ever expanding sph ere in which you suddenly find your self enveloped, and come to observe that from this poi nt extend an infinity of what you perceive as strings of digits, in wha t do appear of random orderings of digits ex tended and branched endlessly in all direc tions by tiny machines that tirelessly append “ digits ” at the end of eac h sequence and spontaneously appear, of nowhere particul ar, to append nine branches from each digit i n each apparent string – “ My God, it’s full of Stars ” (“2001: A Space Odyssey”). You hav e experience d the enumeration of real numbers? Of th e fractional part? Of the na tural numbers? Reality arises of your i deation as to mea ning of what is observed. That i s the problem most essential here. Mat hemati cs stan ds i rref utable on i ts own dry foundations . 1 Which in i ts original text pote ntially infers much more in meaning th rough consensual id eation of wor ds writ. Introduction and Generalities of Import What is presented is premised on basic, common ly kno wn, generally accepted principles for which proofs are readily availabl e in basic theory tex t and taught in basic t heory cour ses at undergraduate l evel. But a few for reader’s refer ence of influence in part on work herein: [8] [ 9] [10]. Fundamental to the “ Continuum Problem ” are noti onal concepts as such exist of our in dividual and co nsensual ideation and representation thereof. From perspective of metaph ys ics, what in our mind we conceptualize a s “ numbers ” do not exist bey ond that discrete real m of our thoughts, whe re such notional concepts (ideations) exist: All else is representation. What it means to be effecti v ely computable was red uced to its essence and put forth by Muḥammad ibn Mū sā al- Khwāriz mī’s in “ The Compendious B ook on Calculation by Completion and B alancing ”, and th e many subsequent works of others: A set of symbols and a set of rules for t heir manipulation. H erein is given the manipulation of sym bols so as to establish a bijection from real numbers into the nat ural nu mbers. Proof of equipollence of real num bers and na tural numbers is accomplished by construct ively establishing a bijection be tween generally accepted representations of natural numbers and real numbers, with the primar y category defining distinction of s aid number s being the fractional part of r eal nu mbers. From the set of symbols S = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } ( digits), togethe r with adjunct set { . } (period) a set of strings is re cursively enumerated. T hat set being simply the Kleene Closure of S , prepended with singular element of { . }; from s uch then is taken that set that is the fr actional parts of real n umbers, as proper s ubset. Let D repres ent that set of string s on S  { . } such that all strings d in D are thus conformin g: d = + = | + = 0 | 1 | 2 | 3 | 4 |5 | 6 | 7 | 8 | 9 By convent ion and generally accept ed theory D mus t contain all pos sible seq uences that can occur as fractional part of any real number that is not an integer. It happens th at D , as defined, will contain sequences that are, by convention and generally acce pted theory, not permissible as th e fractional part of a real number. Such is of no consequence in fact to necessary b ijection, as any subset of a countably-many set is its elf. By elimina tion of impermissible sequences from D set F is obt ained; where F is th at set consistent with the cons ensual human ideation of real numbers. Construction of Set D Let S = { 0, 1, 2 , 3, 4, 5, 6, 7, 8, 9 } be a set o f symbols. Let ‘.’ (period) be a root symbol. Let T be a set of trees, t i , where index i = 0, 1, 2, 3, …, such that: (1) t 0 has only a sing le vertex (bot h root and leaf) labeled with symbol ‘.’, an d; (2) t i is obt ai ned from t i-1 by appending one vertex for each symbol in S to each leaf vertex in t i-1 . Suc h trees be that which our tireless tiny machines recursively enumerated i n endless labor. Each element t i , for i greate r than 0, in T have two important properties: (1) every inner vertex has degre e 10, and; (2) no verte x descendent of any (parent ) vertex has the same labe l as another descendent of th at vertex. Property (2) equivalentl y stated: No tw o siblings of a common parent are labeled with same element of S . Each t ree t i in T ha s finite an d countable number of leaves. Each sequence in each tree of labels from root to any leaf is distinct, b y construction. Each s et of sequences extracted from each tree is itself countable by common and generally accepted theory; in addition to t he cardi nality of each set of such sequences bei ng computable by means within the commonly known and generally accepted theory. All sequences defined by each t i , ta ken collectively as a sin gl e set, constitute the Kleene Closure on S . The set T , indexed by positive integers, is a den umerable set of infin ite cardinality; in which, as elements , ther e exists the representation of sets, ind ivi dually, of fin ite and infinite c ar dinality. Each set its elf denumerable in consequence of havi ng b een recursively en umerated by construction. It is well known and generally accept ed that the un ion of recursively enumerable sets is itself denumer a ble. The set of sequ ences obtained by union of all the sequen ces produced of each t i in T , is by con sequence countable, hen ce denumerabl e. It is of T , by its construction, that D , by its definition, can be obtained in representation; D thus being denumer a ble. It is here that id eation put s b efor e eac h reade r what for many prese nts as an impassable impediment to acceptance: What does it mean that a string is of “ potentially ” i nfinite length. All readers might share a consensual idea tion of “ infinite ”, in conte xt of strings as ex tending without end. So what then makes the sequence (string ) produced by ∑ 3 × 10    of a dif fere nt “infinity” that that p roduced by ∑ 3 × 10    ? Aside from our ideation in regards the former extending to the left of a period and the latter extending to the right in the windmills of our mind with which we tilt? O ur argum ent here, of which e ach reader must in their own mind accept or reject, is that both are composed of “countably many” symbol s and in that fact differ naught in concept of infinity. Infinity is in both cases equivalent. Our constructi o n here of D should make such mathematically and logically irrefutable. Only because we sta ted D w as intended to recursively enumerate all strings that could serve as fractional part of a rea l number do es reader accept s uch. If to ld D r ecursively enumerated the set of string from which all natural numbers can be taken as subset how could what is “ infinite ” change, beyond how each reader con ceives infinite to be in their mind. M athematics i s irrefutab le as to what infinite is, and the hor se sufficiently ridden dead. Of Set D take Set F as Proper Subset Set D is denumerable by constr uction th rough recursive enum eration. It is, similarly, as a set constructed , containing the Kleene Closure on S , and; is composed of count ably-many elements, and; is of “ infinite ” cardinality . D contains F . F b eing that set of all possible sequences that can exist as fractio nal part of any real n umber. From th e mathemati ca l realiti es of D set F as a proper subset o f D is of as infinite cardinality, in any sense, as D . D also contains sequences not permissible as fractional part in our co nsensual ideation of real numbers. D , for instance, contai ns that set of string s containing only the 0 (zero) digit symbol and period in initial position . D also contai ns as sequ ences those that contain no digit save 0 beyond some ordinal position. While no ne of this deni es what is mathemati cally irrefut able a t this point as to factional pa rt of re al nu mbers being equ ipollent to natural numbers, me ans exists to reduc e D to F . Let Z be the set of sequences in D which bey ond some ordinal position only the 0 (zero) symbol exists. Z is therefore that s et of sequences respective t he 0 (zero) sym bol that ar e not allowed as fractional pa rt of any r eal num ber. Set Z , as a p roper sub s et of a countable set, is counta ble. Let F then be obtained by union o f  (empt y sequenc e) and relative compl ement of Z in D (D \ Z) . F thus contains all elements of D except those elements also in Z . F then by construction contains that set of sequences tha t are allowable frac tional part of any real number. F r emains denumer a ble as proper subset of a denumerable set. Of Empirical Objections What Said? By similar exclusion the re ader can further reduc e the set F thus obtained to r emove any strings that offe nd their ide ati on of what is a fra ctional par t of a r eal number to obta in th a t set of strings agreeable to th em. Irre spective what is removed the resulting set r emains countable. I t thus ret ains, where not reduc ed to a finite set, a bijection with th e set of natur al num bers; absent reader’s objecti on that the Kleene Closure on any finite set of symbols pr oduces a “ finite set ” of string s of “ finite length ”. To such reader one might sugge st a re view of category theo ry. By suc h th e set pr oduced by th e closure is of “ countably-many ” (infinite) cardinality. It contains as strings all pos sible strin gs, to include those stri ngs of symb ols that are of “ countab ly-many ” (infinite) ordin al p ositions in length. By consensual representati on of wha t is denoted by countably-many the Kleene Closur e is a set of infinite cardinality, in which there exist sets; of which some such sets are o f infinite cardinality in consequen ce o f being, as well , of infinite ordinal length as represe ntatio n of what is by will idea ti on as strings or seque nces of symbols. Of Thus Enumeration of Real Number s Cantor’s basis for diagonal argument was pre mised on a restriction on F as representation of the set of fract ional parts of real numbers in open interval (0, 1). Such set is by construction given and shown, in contradiction of Cantor’s conclusion to the contrary , to be denum erable – in a bijection placed w ith the set of natural num bers as an infinite subset of an infinite set for which such bij ec tion is proven by rec ursive enumeration of elements. Using Cant or’s “ Pair ing Funct ion ” on th eir Carte sian pr oduct th e sets int egers and F , as obtained above, one obtains a bijection of real numbers with natural numbers; as the Carte sia n product contain s all possibl e representa tions of our ideation of real nu mbers. Pe rhaps Schopenhauer’ s tru e intent was “[r eality] is my representation”? Cantor’s fault of reason exists in Gödel’s subsequent proof as to compl eteness where ta ken in context of rep resentation: A set of hi g her dimensionality canno t be rep resented as a set in lesser dimensiona lity. Cantor’s list being in 2 dimension c o nstructed co uld never contain a set of dimens i onality greater than 2 . Such is in part the essence of Hofstadter’s recursi o n of sets. Elle est la raison There exists then reason giving ri se to necessity ? In the abstra ct cons ider existen ce of a single universal set o f dimension unobservable; that by observat ion appears to contain a finite set o f things. Each element of that set being a single set similarly composed. And so recursively for as long as one might cont inue to examine el ements of any el ement. But in observation one notes that individually each set is not of necessity composed of “s ymbols” from one common se t, as were each set in case of rea l numbers, exc e pt where one at each encounter of a symbol not previously account ed ad ds su ch to t he set of symbols conside red. Stepping back to look again from without one has a recursive construction , from what mi ght be an infinite set of symbols, a “structur e” of what mig ht be infinite dimension s withi n a fin ite constraint. So giving cause to question if in the limit is what is potentially within the structure more n umerable than the real numbers? By co nstruct the an swer must be no. Mirroring the real numbers , up t o the point of real numbers being in each re cursion constrained to a defined finite set of symbo ls, it is o bvious that the struc ture is a “tr ee” growing in potenti ally unbound e d depth and br eadth, and yet at no point, even in the limit, not enumerable. The structure is an abstracti on of any system, a nd abstraction of any system’s prog ressi on in time. It fundamentally gives rise to B ell’s curse of dimensionality in app lication of Dynamic Programming to system s; at the same time giving rise to pote ntial means to overcome Bell’s curse, even if within som e limit. Consequence Recursive enumeration of the set of infi nite cardinality tha t conta ins all strings, of both f inite and infinite o rdinal number of symb ols composed, such th a t each string therein u niquely comprises a represe ntation of a single fractional p art of our c ollective notion of such as part of our representation of such conceptual n otion denoted as “ real numbers ” is establ ished by constructio n. The construction presented makes uses only of well-known and widely accepted mathemati ca l and computational theory. The bijection of real nu mbers with natu ral numbers thus proven is, irr efutably in context of mathematics and logic. Conclusion The set of rea l numbers and set of natural nu mbers are equipollent, as prov e n beyond refutation in mathe matics and logic alone, and; thus g ive in answer of Hilbert’s 1 st problem: No. The hum orous element is that by reason is derived an answer that by comm on knowledge and generally accepted the ory is an effective computation com putable by a Turing Ma chi ne, but one that would ne ver halt so as to render the answer to the operator; save to God alo ne could of such th e answer be obtained, at the end of time. But the n the question di d arise of nowhere particular save of will and repre sentation of the one, and the di ssimilarity of the one ideati on in the minds of th e many. References [1] D. Hilbert, "Mathem atical Problems," Bulletin of New York Mathematical Soci ety, vol. 8, no. 10, pp. 429 -437, 1902. [2] C. Smorymski, Logical Numb er Theory I An Introduction, Springe r -Verlag, 1980, pp. 1 4-21. [3] M. Ma ch ove, Set theory, logic, and th eir limitations, Cambridge University Press, 1996. [4] D. R. Hofstadter , Godel, Escher, Bach: An Eternal Golden Braid, First V inta ge Books, 1 980 ed., Basic Books, Inc, 1979. [5] A. Schopenh auer, The World as Will and Representation, vol. 1 , D over Publications, I nc., 1969. [6] G. Fauconnier, Ma ppings in Though and Language, Cambridge University Press, 1997. [7] J. A. Fodor, Concepts : Where Congitive Science Went Wrong , Oxford University Pr ess, 1998. [8] J. E. M. R. & . U. J. D. Hopcroft , Introduction to Automata Theory, Languages, and Computation , Addison Wesley, 1979. [9] D. C. Kozen, Aut omata and Computa bili ty, Sprin ger, 1997. [10] D. C. Kozen, Theory o f Computation, Springe r, 2006. Contributions & Acknowledgme nt Fiorella Marincic h-Sauerbier – Cont ribution to ke y and critical c onceptual points no t rising to commonly accepte d stand ard for authorship; though invaluable in work p resented and efforts to redu ce t hought t o (hopefully) cogent representation in a human comprehens i ble language.