3 research outputs found
Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?
The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part
Abstraktnà studium úplnosti pro infinitárnà logiky
V tĂ©to dizertaÄŤnĂ práci se zabĂ˝váme studiem vlastnostĂ Ăşplnosti infinitárnĂch vĂ˝rokovĂ˝ch logik z pohledu abstraktnĂ algebraickĂ© logiky. CĂlem práce je pochopit, jak lze základnĂ nástroj v dĹŻkazech uplnosti, tzv. Lindenbaumovo lemma, zobecnit za hranici finitárnĂch logik. Za tĂmto účelem studujeme vlastnosti Ăşzce souvisejĂcĂ s LindenbaumovĂ˝m lemmatem (a v dĹŻsledku takĂ© s vlastnostmi Ăşplnosti). UvidĂme, Ĺľe na základÄ› tÄ›chto vlastnostĂ lze vystavÄ›t novou hierarchii infinitárnĂch vĂ˝rokovĂ˝ch logik. TakĂ© se zabĂ˝váme studiem tÄ›chto vlastnostĂ v pĹ™ĂpadÄ›, kdy naše logika má nÄ›jakĂ© (pĹ™ĂpadnÄ› hodnÄ› obecnÄ› definovanĂ©) spojky implikace, disjunkce a negace. Mimo jinĂ© uvidĂme, Ĺľe pĹ™Ătomnost danĂ˝ch spojek mĹŻĹľe zajist platnost Lindenbaumova lemmatu. Keywords: abstraktnĂ algebraická logika, infinitárnĂ logiky, Lindenbau- movo lemma, disjunkce, implikace, negaceIn this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negationKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult