Gentzen’s consistency proof without heightlines

2013 ◽  
Vol 52 (3-4) ◽  
pp. 449-468 ◽  
Author(s):  
Annika Siders
Keyword(s):  
Consistency Proof

scholarly journals Formalism and Hilbert’s understanding of consistency problems

2021 ◽  
Author(s):  
Michael Detlefsen
Keyword(s):  
Philosophy Of Mathematics ◽  
Model Construction ◽  
Traditional Model ◽  
Consistency Proof ◽  
Salient Points ◽  
Consistency Problem ◽  
Late Nineteenth ◽  
Similarity And Difference ◽  
Traditional Understanding

AbstractFormalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism (advocated by Heine), game formalism (advocated by Thomae) and instrumental formalism (advocated by Hilbert). After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. My primary aim there will be to develop its formalist elements more fully. These are, in the main, (i) its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, (ii) its departure from the traditional understanding of the basic nature of proof and (iii) its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all.

scholarly journals An Investigation on the Logical Structure of Mathematics (VI): Consistent V-System T(V) (With Corrections to Part (XII))

1959 ◽  
Vol 14 ◽  
pp. 95-107
Author(s):  
Sigekatu Kuroda
Keyword(s):  
Logical Structure ◽  
Consistency Proof ◽  
Truth Values ◽  
Definition Of

The V-system T(V) is defined in §2 by using §1, and its consistency is proved in §3. The definition of T(V) is given in such a way that the consistency proof of T(V) in §3 shows a typical way to prove the consistency of some subsystems of UL. Otherwise we could define T(V) more simply by using truth values. After T(V)-sets are treated in §4, it is proved in §5 as a T(V)-theorem that T(V)-sets are all equal to V.

A weak absolute consistency proof for some systems of illative combinatory logic

1983 ◽  
Vol 48 (3) ◽  
pp. 771-776 ◽  
Author(s):  
M.W. Bunder
Keyword(s):  
Set Theory ◽  
Combinatory Logic ◽  
Weak Consistency ◽  
Consistency Proof ◽  
First Order ◽  
Formal Systems ◽  
Elementary Methods ◽  
Order Arithmetic ◽  
Strongly Consistent ◽  
Order Predicate Calculus

A large number of formal systems based on combinatory logic or λ-calculus have been extended to include first order predicate calculus. Several of these however have been shown to be inconsistent, all, as far as the author knows, in the strong sense that all well formed formulas (which here include all strings of symbols) are provable. We will call the corresponding consistency notion—an arbitrary wff ⊥ is provable—weak consistency. We will say that a system is strongly consistent if no formula and its negation are provable.Now for some systems, such as that of Kuzichev [11], the strong and weak consistency notions are equivalent, but in the systems of [5] and [6], which we will be considering, they are not. Each of these systems is strong enough to have all of ZF set theory, except Grounding and Choice, interpretable in it, and the system of [5] can also encompass first order arithmetic (see [7]). It therefore seems unlikely that a strong consistency result could be proved for these systems using elementary methods. In this paper however, we prove the weak consistency of both these systems by means that could be formulated, at least within the theory of [5]. The method also applies to the typed systems of Curry, Hindley and Seldin [10] and to Seldin's generalised types [12].

Towards a consistency proof for immiscible lattice BGK

PAMM ◽  
2003 ◽  
Vol 3 (1) ◽  
pp. 80-83
Author(s):  
Dirk Kehrwald
Keyword(s):  
Consistency Proof

Analysis without actual infinity

1981 ◽  
Vol 46 (3) ◽  
pp. 625-633 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Natural Science ◽  
Order Theory ◽  
Finite Part ◽  
Consistency Proof ◽  
Actual Infinity ◽  
First Order ◽  
Finite Models ◽  
First Order Theory ◽  
Potential Applications

AbstractWe define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.

scholarly journals ERNA and Friedman's Reverse Mathematics

2011 ◽  
Vol 76 (2) ◽  
pp. 637-664 ◽  
Author(s):  
Sam Sanders
Keyword(s):  
Nonstandard Analysis ◽  
Reverse Mathematics ◽  
Consistency Proof ◽  
Transfer Principles ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractElementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA0 from Reverse Mathematics (see [21] and [22]) can be ‘pushed down’ into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality ‘up to infinitesimals’. It turns out that ERNA plays the role of RCA0 and that transfer for universal formulas corresponds to WKL.

Sign in / Sign up

Export Citation Format

Share Document