Title
Gedel o aksiomatizaciji teorije skupova
Creator
Adžić, Miloš R., 1982-
Copyright date
2014
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Autorstvo 3.0 Srbija (CC BY 3.0)
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Language
Serbian
Cobiss-ID
Theses Type
Doktorska disertacija
description
Datum odbrane: 26.12.2014.
Other responsibilities
mentor
Došen, Kosta, 1954-
član komisije
Lazović, Živan, 1958-
član komisije
Todorčević, Stevo
član komisije
Perović, Aleksandar
Academic Expertise
Društveno-humanističke nauke
University
Univerzitet u Beogradu
Faculty
Filozofski fakultet
Group
Odeljenje za filozofiju
Alternative title
Gödel on axiomatization of set theory
Publisher
[M. R. Adžić]
Format
279 listova
description
Filozofija - Filozofija matematike / Philosophy - Philosophy of mathematics
Abstract (sr)
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
Abstract (en)
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.
Authors Key words
Kurt Gedel, platonizam, teorija skupova, nove aksiome, refleksija
Authors Key words
Kurt Gödel, platonism, set theory, new axioms, reflection
Type
Tekst
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