Search results for "Gödel"
showing 10 items of 18 documents
What is Mathematics: Gödel's Theorem and Around (Edition 2013)
2013
Hyper-textbook for students in mathematical logic and foundations of mathematics. Edition 2013. ATTENTION! New Edition 2015 available at https://dspace.lu.lv/dspace/handle/7/5306.
Protoalgebraicity and the Deduction Theorem
2001
This chapter is intended as an introduction to the Deduction Theorem and to applications of this theorem in metalogic.
Phenomenological-Semantic Investigations into Incompleteness
2000
When today the phenomenologist surveys the history of the philosophical comprehension of Godel’s theorems, he is confronted with the realization that the decisive publications come almost exclusively from the sphere of analytic philosophy.1 But does phenomenology in the spirit of Husserl not mean to keep in step with the epochal results of the special sciences by working on the phenomenological understanding of them? Phenomenological research of this kind means the same as development of phenomenological theory of science (Wissenschaftstheorie). In connection with the incompleteness theorems, the latter would be confronted with fundamental questions such as, “To what extent can mathematical…
The mathematics-physics analogy
1995
The basic goal of this chapter is to delve deeply into Godel’s second great strategy in the philosophy of mathematics: the analogy between deductive and empirical sciences. Moreover, I shall try to explore the holistic, and even conventionalist implications of the analogy, such as it appears in some contemporary philosophers of mathematics who have defended the analogy to some extent. To do this, it has been necessary to present an overview of the most important precedents in the use of the analogy, such as Russell, Hilbert, Carnap, Tarski, Quine. After that, I shall present Godel’s views on the analogy, in both his published and unpublished writings. Surprisingly, most of these authors mai…
Logic, Computing and Biology
2015
Logic and Computing are appropriate formal languages for Biology, and we may well be surprised by the strong analogy between software and DNA, and between hardware and the protein machinery of the cell. This chapter examines to what extent any biological entity can be described by an algorithm and, therefore, whether the Turing machine and the halting problem concepts apply. Last of all, I introduce the concepts of recursion and algorithmic complexity, both from the field of computer science, which can help us understand and conceptualise biological complexity.
Gödelin epätäydellisyyslauseet
2016
Ossi Kosonen, Gödelin epätäydellisyyslauseet, Gödel's incompleteness , matematiikan pro gradu -tutkielma, 57 sivua, Jyväskylän yliopisto, Matematiikan ja tilastotieteen laitos, syksy 2016. Tämän tutkielman tarkoituksena on todistaa Gödelin kaksi epätäydellisyyslausetta RA-kielessä. Itävaltalais-amerikkalainen Kurt Gödel todisti nimeänsä kantavat lauseet artikkelissaan vuonna 1931. Gödel ei itse varsinaisesti käyttänyt RA-kieltä lauseiden alkuperäisissä todistuksissa, mutta tässä tutkielmassa RA-kieli on valittu formaaliksi kieleksi, koska se perustuu predikaattikielten pohjalle. RA-kielen tarkoitus on formalisoida mahdollisimman hyvin aritmetiikka, joka käytännössä onnistuu mallintamalla lu…
Kolmogorov numberings and minimal identification
1995
Identification of programs for computable functions from their graphs by algorithmic devices is a well studied problem in learning theory. Freivalds and Chen consider identification of ‘minimal’ and ‘nearly minimal’ programs for functions from their graphs. To address certain problems in minimal identification for Godel numberings, Freivalds later considered minimal identification in Kolmogorov Numberings. Kolmogorov numberings are in some sense optimal numberings and have some nice properties. We prove certain hierarchy results for minimal identification in every Kolmogorov numbering. In addition we also compare minimal identification in Godel numbering versus minimal identification in Kol…
The character and origin of the manuscripts in the present edition
1995
In the following I try to provide the reader with all the details needed to insert the Godel essays appearing here in the broader context of the rest of his unpublished work, as well as everything about my personal work on them. I start with a short description of what can be found in the Godel Nachlass in Princeton, USA, by referring to the catalogue prepared by John Dawson. Then I attempt to justify the particular selection I made of the manuscripts which I finally decided to study, reconstruct and publish. Such an explanation seems to be advisable given the great amount of Godel’s material unpublished but still extant. Also, I describe some of the historical details relevant to understan…
The analytic-synthetic distinction
1995
This chapter tries to throw light on the first of Godel’s two main theses in the philosophy of mathematics, namely that mathematical propositions are analytic. To this end, an overview of similar conceptions is presented first in which the views by Frege, Russell Wittgenstein, Carnap and Quine are expounded. Then Godel’s view is analyzed, both in his publications and in the manuscripts which appear in this edition. The presentation of Carnap’s detailed attempt to define analyticity in his The Logical Syntax of Language (1934) may seem rather long in comparison with the ones devoted to the other authors, but it should be recalled that the Godel manuscripts appearing here were a direct philos…
Realism, metamathematics, and the unpublished essays
1995
This initial chapter is divided into two sections. The first is devoted to a brief exposition of the intuitive essence and the philosophical motivation of Godel’s main metamathematical results, namely his completeness theorem for elementary logic (1930) and his incompleteness theorems for arithmetic (1931). Thereafter some discussion of the different ways to confront the relationship between those results and Godel’s philosophical realism in logic and mathematics is offered. Thus, mathematical realism will be successively regarded as (i) a philosophical consequence of those results; (ii) a heuristic principle which leads to them; (iii) a philosophical hypothesis which is “verified” by them.…