Naivna teorija skupova

Naivna teorija skupova je teorija skupova u kojoj su skupovi uvedeni koristeći tzv. samoevidentni koncept skupova kao kolekcija objekata smatranih celinom. Ona predstavlja početnu fazu u izgradnji teorije skupova, i obuhvata vreme kad je njen osnivač Georg Kantor objavio radove o teoriji skupova 1871. godine do pojave prvih paradoksa. On se pri izradi nije služio aksiomima, ali su sve teoreme koje je dobio izvodive iz tri aksioma: ekstenzionalnosti, komprehenzije i izbora.

U toj teoriji skup je primitivan pojam koji se kao takav ne defini]e. Podrazumeva se da čilac već ima izgrađenu intuiciju o pojmu skupa, odnosno da je skup kolekcija objekata koji zajedno čine celinu. Veliki deo teorije skupova Kantor je izgradio na ovakvom nedefiniranom i vrlo nejasnom pojmu skupa. Konsekventno, kad je teorija postala priznata, pojavili su se paradoksi. Pojave paradoksa (koji su se pojavili Raselovim otkrićem paradoksa) i nerešivih problema u naivnoj teoriji izbjegavane su uvođenjem teorije tipova, teorije klasa i dr. Slabe osnove pokazale su potrebu za aksiomima i Ernst Zermelo je 1908. godine predložio aksiomatizaciju teorije, dokazavši da se može dobro urediti svaki skup. Uvođenjem aksioma teorija se razvila, te se nastala aksiomatska teorija skupova.

Skupovi su od velike važnosti u matematici; u modernim formalnim tretmanima većina matematičkih objekata (brojevi, odnosi, funkcije, itd.) su definisani u smislu skupova. Naivna teorija skupova je dovoljna za mnoge svrhe, a ujedno služi i kao odskočna daska ka formalnijim tretmanima.

Metod

Naïvna teorija u smislu „naivne teorije skupova” je neformalizovana teorija, odnosno teorija koja koristi prirodni jezik za opisivanje skupova i operacija na skupovima. Reči i, ili, ako ... onda, ne, za neke, za svaki se tretiraju kao u običnoj matematici. Kao pogodnost, upotreba naivne teorije skupova i njen formalizam preovlađuju čak i u višoj matematici - uključujući i formalnije postavke same teorije skupova.

Prvi razvoj teorije skupova bila je naivna teorija skupova. Nju je kreirao krajem 19. veka Georg Kantor kao deo njegove studije beskonačnih skupova, a razvio ju je Gotlob Frege u svom radu Begriffsschrift.

Naivna teorija skupova se može odnositi na nekoliko vrlo različitih pojmova. To može biti

• Neformalni prikaz teorije aksiomatičnih skupova, npr. kao u Naivnoj teoriji skupova, Pola Halmosa.
• Rane ili kasnije verzije Georg Kantorove teorije i drugih neformalnih sistema.
• Odlučno nedosledne teorije (bilo aksiomatske ili ne), kao što je teorija Gotloba Grege koja je proizvela Raselov paradoks, te teorije Đuzepe Peana i Ričarda Dedekinda.

Paradoksi

Pretpostavka da se bilo koje svojstvo može koristiti za formiranje skupa, bez ograničenja, dovodi do paradoksa. Jedan čest primer je Raselov paradoks: ne postoji skup koji se sastoji od „svih skupova koji ne sadrže sebe”. Stoga dosledni sistemi naivne teorije skupova moraju da sadrže i neka ograničenja u principima koji se mogu koristiti za formiranje skupova.

Kantorova teorija

Neki smatraju da Georg Kantorova teorija skupova zapravo nije bila umešana u skupovno-teoretske paradokse (pogledajte Frápolli 1991). Jedna od poteškoća da se to sa sigurnošću utvrdi je što Kantor nije pružio aksiomatizaciju svog sistema. Do 1899. godine, Kantor je bio svestan nekih paradoksa proizašlih iz neograničenog tumačenja njegove teorije, na primer, Kantorovog paradoksa i Barali-Fortijevog paradoksa, i nije smatrao da su oni diskreditovali njegovu teoriju. Kantorov paradoks može se izvesti iz gornje (pogrešne) pretpostavke - da se svako svojstvo P(x) može koristiti za formiranje skupa - uzimajući da je P(x) x kardinalni broj. Frege je eksplicitno aksiomatizovao teoriju u kojoj se može interpretirati formalizovana verzija naivne teorije skupova, i upravo je na tu formalnu teoriju Bertrand Rasel referirao kada je izneo svoj paradoks, a ne nužno Kantorovu teoriju.

Aksiomatske teorije

Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when.

Consistency

A naive set theory is not necessarily inconsistent, if it correctly specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms. It is possible to state all the axioms explicitly, as in the case of Halmos' Naive Set Theory, which is actually an informal presentation of the usual axiomatic Zermelo–Fraenkel set theory. It is "naive" in that the language and notations are those of ordinary informal mathematics, and in that it does not deal with consistency or completeness of the axiom system.

Likewise, an axiomatic set theory is not necessarily consistent: not necessarily free of paradoxes. It follows from Gödel's incompleteness theorems that a sufficiently complicated first order logic system (which includes most common axiomatic set theories) cannot be proved consistent from within the theory itself – even if it actually is consistent. However, the common axiomatic systems are generally believed to be consistent; by their axioms they do exclude some paradoxes, like Russell's paradox. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at all in these theories or in any first-order set theory.

The term naive set theory is still today also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory.

Utility

The choice between an axiomatic approach and other approaches is largely a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory. References to particular axioms typically then occur only when demanded by tradition, e.g. the axiom of choice is often mentioned when used. Likewise, formal proofs occur only when warranted by exceptional circumstances. This informal usage of axiomatic set theory can have (depending on notation) precisely the appearance of naive set theory as outlined below. It is considerably easier to read and write (in the formulation of most statements, proofs, and lines of discussion) and is less error-prone than a strictly formal approach.

Skupovi, članstvo i jednakost

In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.

Passage with the original set definition of Georg Cantor

The definition of sets goes back to Georg Cantor. He wrote in his 1915 article:

“Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.” – Georg Cantor

“A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.” – Georg Cantor

Literatura

• Bourbaki, N., Elements of the History of Mathematics, John Meldrum (trans.), Springer-Verlag, Berlin, Germany, 1994.
• Cantor, Georg (1874), „Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen”, J. Reine Angew. Math., 77: 258—262, doi:10.1515/crll.1874.77.258, See also pdf version:  
• María J. Frápolli|Frápolli, María J., 1991, "Is Cantorian set theory an iterative conception of set?". Modern Logic, v. 1 n. 4, 1991, 302–318.
• Frege, Gottlob (1893), Grundgesetze der Arithmetik, 1, Jena 1893. 
• Halmos, Paul, Naïve Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2. 
• Kelley, J.L., General Topology, Van Nostrand Reinhold, New York, NY, 1955.
• van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. ISBN 0-674-32449-8.
• Meschkowski, Herbert; Nilson, Winfried (1991), Georg Cantor: Briefe. Edited by the authors., Berlin: Springer, ISBN 3-540-50621-7 
• Peano, Giuseppe (1889), Arithmetices Principies nova Methoda exposita, Turin 1889. 
• Zermelo, Ernst (1932), Georg Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind. Edited by the author., Berlin: Springer 
• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN 0-444-85401-0 
• Johnson, Philip (1972), A History of Set Theory, Prindle, Weber & Schmidt, ISBN 0-87150-154-6 
• Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd изд.), Springer Verlag, ISBN 0-387-94094-4, doi:10.1007/978-1-4612-0903-4 
• Ferreirós, Jose (2001), Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Berlin: Springer, ISBN 978-3-7643-5749-8 
• Monk, J. Donald (1969), Introduction to Set Theory, McGraw-Hill Book Company, ISBN 978-0-898-74006-6 
• Potter, Michael (2004), Set Theory and Its Philosophy: A Critical Introduction, Oxford University Press, ISBN 978-0-191-55643-2 
• Smullyan, Raymond M.; Fitting, Melvin (2010), Set Theory and the Continuum Problem, Dover Publications, ISBN 978-0-486-47484-7 
• Tiles, Mary (2004), The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise, Dover Publications, ISBN 978-0-486-43520-6 
• Rudin, Walter B. (6. 4. 1990). „Set Theory: An Offspring of Analysis”. Marden Lecture in Mathematics. University of Wisconsin-Milwaukee. Архивирано из оригинала 2021-10-31. г. — преко YouTube. CS1 одржавање: Формат датума (веза)

Spoljašnje veze

• Beginnings of set theory page at St. Andrews
• Earliest Known Uses of Some of the Words of Mathematics (S)
• Daniel Cunningham, Set Theory article in the Internet Encyclopedia of Philosophy.
• Jose Ferreiros, "The Early Development of Set Theory" article in the [Stanford Encyclopedia of Philosophy].
• Foreman, Matthew, Akihiro Kanamori, eds. Handbook of Set Theory. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993).
• Hazewinkel Michiel, ур. (2001). „Axiomatic set theory”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Hazewinkel Michiel, ур. (2001). „Set theory”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Schoenflies, Arthur (1898). Mengenlehre in Klein's encyclopedia.
• Online books, and library resources in your library and in other libraries about set theory