Another Splitting of the Pasch Axiom

2011 ◽  
Vol 59 (3-4) ◽  
pp. 219-227 ◽  
Author(s):  
Victor Pambuccian
Keyword(s):  
Pasch Axiom ◽  
The Pasch Axiom

Splitting the Pasch axiom

1996 ◽  
Vol 56 (1-2) ◽  
pp. 126-130 ◽  
Author(s):  
Victor Pambuccian
Keyword(s):  
Pasch Axiom ◽  
The Pasch Axiom

scholarly journals Prolegomena to any theory of proof simplicity

2019 ◽  
Vol 377 (2140) ◽  
pp. 20180035 ◽  
Author(s):  
Victor Pambuccian
Keyword(s):  
Simple Proof ◽  
Axiom System ◽  
Elementary Geometry ◽  
Theme Issue ◽  
Pasch Axiom ◽  
Axiom Systems ◽  
The Pasch Axiom

By looking at concrete examples from elementary geometry, we analyse the manner in which the simplicity of proofs could be defined. We first find that, when presented with two proofs coming from mutually incompatible sets of assumptions, the decision regarding which one is simplest can be made, if at all, only on the basis of reasoning outside of the formal aspects of the axiom systems involved. We then show that, if the axiom system is fixed, a measure of proof simplicity can be defined based on the number of uses of axioms deemed to be deep or valuable, and prove a number of new results regarding the need to use at least three times some axioms in the proof of others. One such major example is Pappus implies Desargues, which is shown to require three uses of Pappus. A similar situation is encountered with Veblen's proof that the outer form of the Pasch axiom implies the inner form thereof. The outer form needs to be used at least three times in any such proof. We also mention the likely conflicting requirements of directness of a proof and the length of a proof. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

Forms of the Pasch axiom in ordered geometry

2010 ◽  
Vol 56 (1) ◽  
pp. 29-34 ◽  
Author(s):  
Victor Pambuccian
Keyword(s):  
Pasch Axiom ◽  
The Pasch Axiom

Determinateness and the Pasch Axiom

1973 ◽  
Vol 16 (2) ◽  
pp. 159-160 ◽  
Author(s):  
Andrew Adler
Keyword(s):  
Essential Role ◽  
Euclidean Geometry ◽  
Axiom Of Choice ◽  
Hamel Basis ◽  
Order Continuity ◽  
Full List ◽  
Pasch Axiom ◽  
The Axiom Of Choice ◽  
The Pasch Axiom ◽  
Continuity Axiom

Let E be the (nonelementary) plane Euclidean geometry without the Pasch axiom. (The Pasch axiom says that a line cutting one side of a triangle must also cut another side. A full list of axoms for E is given in [5].) E satisfies in particular the full second-order continuity axiom.Szczerba [5] has recently shown using a Hamel basis for the reals over the rationals that there exists a model of E not satisfying the Pasch axiom. It is natural to ask whether the axiom of choice plays an essential role in the proof. It will turn out that it does.

The Induced Path Transit Function and the Pasch Axiom

2015 ◽  
Vol 39 (S1) ◽  
pp. 123-134 ◽  
Author(s):  
Manoj Changat ◽  
Iztok Peterin ◽  
Abisha Ramachandran ◽  
Aleksandra Tepeh
Keyword(s):  
Pasch Axiom ◽  
Induced Path ◽  
The Pasch Axiom
Sign in / Sign up

Export Citation Format

Share Document