Search results for "Gödel"

Lambda substitution algebras

1993

In the paper an algebraic metatheory of type-free λ-calculus is developed. Our version is based on lambda substitution algebras (λSAs), which are just SAs introduced by Feldman (for algebraizing equational logic) enriched with a countable family of unary operations of λ-abstraction and a binary operation of application. Two representation theorems, syntactical and semantic, are proved, what directly provides completeness theorems.

Gödel and the Blind Watchmaker

2015

While accepting that contingency is key to biological evolution, we wonder how much need there is for it. It is extremely difficult to talk about trends in evolution, but the fact remains that they are found here and there when evolutionary experiments are repeated. But we should ask, for example, whether there is an unavoidable tendency of life towards progressive complexity . This chapter deals with certain theoretical considerations from Logic and Computing on the conditions necessary to formulate a predictive evolutionary theory .

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…

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…

Essays on Gödel's Reception of Leibniz, Husserl, and Brouwer

2015

Performability of Actions

2021

AbstractAction theory may be regarded as a theoretical foundation of AI, because it provides in a logically coherent way the principles of performing actions by agents. But, more importantly, action theory offers a formal ontology mainly based on set-theoretic constructs. This ontology isolates various types of actions as structured entities: atomic, sequential, compound, ordered, situational actions etc., and it is a solid and non-removable foundation of any rational activity. The paper is mainly concerned with a bunch of issues centered around the notion of performability of actions. It seems that the problem of performability of actions, though of basic importance for purely practical ap…

Leon Henkin the Reviewer

2014

In this chapter, we intend to look at Henkin’s reviews, a total of forty-six. The books and papers reviewed deal with a large variety of subjects that range from the algebraic treatment of logical systems to issues concerning the philosophy of mathematics and, not surprisingly—given his active work in mathematical education—one on the teaching of this subject. Most of them were published in The Journal of Symbolic Logic and only one in the Bulletin of the American Mathematical Society. We will start by sorting these works into subjects and continue by providing a brief summary of each of them in order to point out those aspects that are originally from Henkin, and what we take to be mistake…

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.…

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…

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…