Gödel on axiomatization of set theory

[M. R. Adžić]

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Filozofija - Filozofija matematike / Philosophy - Philosophy of mathematics

Cilj ovog rada je da ispita detaljeGedelove (Kurt Gödel) platonistiˇcke pozicije u filozofiji matematike, argumente koji se iznose za i protiv nje, kao i posledice koje ova pozicija ima za formulisanje novih aksioma teorije skupova. Razmotri´cemo neke konkretne predloge novih aksioma koje je Gedel ponudio, pre svega jake aksiome beskonaˇcnosti ili aksiome velikih kardinala, kao i Gedelovu sugestiju da centralni princip na kojem bi trebalo da poˇcivaju sve nove aksiome teorije skupova jeste princip refleksije. Osim toga, uporedi´cemo Gedelova gledišta o novim aksiomama sa nekim savremenimgledištima da bismo videli da li se i u kojoj meri ova poslednjamogu smatrati unapre ¯ denjima Gedelovog programa. Prva glava ovog rada je uvodnog karaktera i ne donosi ništa novo. U njoj ´cemo da uvedemo pojmove i rezultate na koje ´cemo se ˇcesto oslanjati u nastavku rada. Poslednji deo ovog uvoda sadrži nešto detaljniji pregled ˇcitavog rada nego što smomogli da pružimo u ovom rezimeu. Glave 2 i 3 posve´cene su detaljima Gedelovog platonistiˇckog stanovišta, glavnim kritikama koje su mu upu´civane i nekimmogu´cim odgovorima na njih. Posebno ´ce nas zanimati Gedelov pojam intuicije koji je bio meta mnogih kritika. Pokaza´cemo da se ovaj pojam ne treba da razume kao "misti ˇcna sposobnost neposrednog uvida" u strukturu apstraktnog matemati ˇckog sveta, i da u Gedelovim radovima do 1959. godine ova sposobnost predstavlja ništa drugo nego ˇcin razumevanja osnovnih matematiˇckih pojmova. Slede´ca glava, ˇcetvrta po redu, posve´cena je analizi dve vrste opravdanja novih aksioma, onako kako ih je Gedel razumeo. Reˇc je o unutrašnjem i spoljašnjem opravdanju. Vide´cemo da nijednoj od ovih vrsta opravdanja Gedel ne daje preimu´cstvo, kako se to ˇcesto smatra, i da izme ¯ du njih postoji povratni odnos koji ´cemo da objasnimo u ˇcetvrtom odeljku ove glave. U glavi 5 ispita´cemo jedan po našem mišljenju neuspešan argument u prilog matematiˇckom platonizmu koji delimiˇcno poˇciva na pogrešnom razumevanju Gedelovog stanovišta. Šesta glava posve´cena je Gedelovoj sugestiji da je princip refleksije centralan za teoriju skupova i da bi u izvesnom smislu svi kandidati za nove aksiome trebalo na njemu da poˇcivaju. Ispita´cemo ovaj princip u razliˇcitim kontekstima i videti da ako se on razume u odre ¯ denomsmislu, onda je svaka njegova do danas poznata formulacija preslaba da uˇcini ono što se od nje oˇcekuje. Poslednja, sedma glava ovog rada posve´cena je nekim savremenim gledištima formulisanja novih aksioma teorije skupova i njihovoj filozofskoj pozadini

This aim of this work is to investigate Kurt Gödel’s platonistic position in the philosophy of mathematics, arguments up in its favor or against it, as well as the consequences this position has for the formulation of new axioms of set theory. We shall examine some particular proposals for new axioms made by Gödel - most notably strong axioms of infinity or large cardinal axioms, together with Gödel’s suggestion that the crucial principle upon which all new axioms of set theory should be founded is the principle of reflection. Apart from that, we shall compare Gödel’s views concerning new axiomswith some recent developments in order to appreciatewhether the later can be seen as continuations of Gödel’s program. Chapter 1 presents an introduction and is not original. Here we introduce the notions and results which we shall need in the rest of the work. The last part of this introduction offers a somewhat more detailed summary of this work then we were able to offer here. Chapters 2 and 3 focus on the details of Gödel’s platonistic standpoint, main forms of criticisms that were marshaled against, it as well as some possible replies to these. We shall be concerned in particular with Gödel’s concept of intuition, which proved to be in the center of most critical attacks. We shall attempt to showthat this concept should not be understood as the “mystical faculty of direct insight” into the structure of the abstract mathematical world, and that in Gödel’s works prior to 1959 this faculty amounts to nothing more then the act of understanding basic mathematical concepts. Chapter 4 presents an analysis of two types of justification of new axioms as envisaged byGödel, namely intrinsic and extrinsic justification. We shall see that Gödel does not ascribe primacy to either of these two types of justification, as it is often thought, and that there is a certain “back and forth” interaction between them which we shall explain in the fourth section of this chapter. In Chapter 5 we shall critically examine what we find to be an unsuccessful argument for mathematical platonism, which partly rests on a misunderstanding of Gödel’s standpoint. The following chapter, Chapter 6, starts with Gödel’s suggestion that reflection principle is of prime importance for set theory, and that in a certain sense all candidates for new axioms should be founded upon it. We shall examine this principle in various contexts and show that if the principle is understood in a certain sense, then each of its various formulations proposed until now is too weak to deliver what is expected of it. The final chapter of this work, Chapter 7, provides an examination of some recent attempts at formulating new axioms for set theory, as well as their philosophical background.

Kurt Gedel, platonizam, teorija skupova, nove aksiome, refleksija

Kurt Gödel, platonism, set theory, new axioms, reflection

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