1,736 research outputs found
Tarski Geometry Axioms – Part II
In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation),of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1),congruence equivalence relation (A2),congruence identity (A3),segment construction (A4),SAS (A5),betweenness identity (A6),Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8),the upper dimension axiom (A9),the Euclid axiom (A10),the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS).In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes.Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].Coghetto Roland - Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrabowski Adam - Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandCzesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99–107, 2005.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Roland Coghetto. Circumcenter, circumcircle and centroid of a triangle. Formalized Mathematics, 24(1):17–26, 2016. doi:10.1515/forma-2016-0002.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Adam Grabowski. Efficient rough set theory merging. Fundamenta Informaticae, 135(4): 371–385, 2014. doi:10.3233/FI-2014-1129.Adam Grabowski. Tarski’s geometry modelled in Mizar computerized proof assistant. In Proceedings of the 2016 Federated Conference on Computer Science and Information Systems, FedCSIS 2016, Gdańsk, Poland, September 11–14, 2016, pages 373–381, 2016. doi:10.15439/2016F290.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211–221, 2015. doi:10.1007/s10817-015-9333-5.Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5–10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300–314. Springer, 2010. doi:10.1007/978-3-642-14128-7_26.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. 2012. Master’s thesis.Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In F. Botana and T. Recio, editors, Automated Deduction in Geometry, volume 4869 of Lecture Notes in Computer Science, pages 139–156. Springer, 2007.William Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics, 22(2):167–176, 2014. doi:10.2478/forma-2014-0017.Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.Alfred Tarski and Steven Givant. Tarski’s system of geometry. Bulletin of Symbolic Logic, 5(2):175–214, 1999.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.Wojciech A. Trybulec. Axioms of incidence. Formalized Mathematics, 1(1):205–213, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990
Proof-checking Euclid
We used computer proof-checking methods to verify the correctness of our
proofs of the propositions in Euclid Book I. We used axioms as close as
possible to those of Euclid, in a language closely related to that used in
Tarski's formal geometry. We used proofs as close as possible to those given by
Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48
propositions, we proved 235 theorems. The extras were partly "Book Zero",
preliminaries of a very fundamental nature, partly propositions that Euclid
omitted but were used implicitly, partly advanced theorems that we found
necessary to fill Euclid's gaps, and partly just variants of Euclid's
propositions. We wrote these proofs in a simple fragment of first-order logic
corresponding to Euclid's logic, debugged them using a custom software tool,
and then checked them in the well-known and trusted proof checkers HOL Light
and Coq.Comment: 53 page
Vienna Circle and Logical Analysis of Relativity Theory
In this paper we present some of our school's results in the area of building
up relativity theory (RT) as a hierarchy of theories in the sense of logic. We
use plain first-order logic (FOL) as in the foundation of mathematics (FOM) and
we build on experience gained in FOM.
The main aims of our school are the following: We want to base the theory on
simple, unambiguous axioms with clear meanings. It should be absolutely
understandable for any reader what the axioms say and the reader can decide
about each axiom whether he likes it. The theory should be built up from these
axioms in a straightforward, logical manner. We want to provide an analysis of
the logical structure of the theory. We investigate which axioms are needed for
which predictions of RT. We want to make RT more transparent logically, easier
to understand, easier to change, modular, and easier to teach. We want to
obtain deeper understanding of RT.
Our work can be considered as a case-study showing that the Vienna Circle's
(VC) approach to doing science is workable and fruitful when performed with
using the insights and tools of mathematical logic acquired since its formation
years at the very time of the VC activity. We think that logical positivism was
based on the insight and anticipation of what mathematical logic is capable
when elaborated to some depth. Logical positivism, in great part represented by
VC, influenced and took part in the birth of modern mathematical logic. The
members of VC were brave forerunners and pioneers.Comment: 25 pages, 1 firgure
On non-measurable sets and invariant tori
The question: "How many different trajectories are there on a single
invariant torus within the phase space of an integrable Hamiltonian system?" is
posed. A rigorous answer to the question is found both for the rational and the
irrational tori. The relevant notion of non-measurable sets is discussed.Comment: 8 pages, 1 figur
A logic road from special relativity to general relativity
We present a streamlined axiom system of special relativity in first-order
logic. From this axiom system we "derive" an axiom system of general relativity
in two natural steps. We will also see how the axioms of special relativity
transform into those of general relativity. This way we hope to make general
relativity more accessible for the non-specialist
The axiom of choice and the paradoxes of the sphere.
Thesis (M.A.)--Boston UniversityThe Axiom of Choice is stated in the following form: For every set Z whose elements are sets A, non-empty and mutually disjoint, there exists at least one set B having one and only one element from each of the sets A belonging to Z. Examples are given to show the use of the Axiom of Choice and also to show when it is not needed.
Two other fundamental terms are defined, namely "congruence" and "equivalence by finite decomposition", and examples are given. Congruence is defined as follows: The sets of points A and B are congruent: A B, if there exists a function f, which transforms A into B in a one-to-one manner such that if a1 and a2 are two arbitrary points of the set A, then d(a1, a2)=d[f(a1), f(a2)]; d(a, b) is a real number called the distance between the points a and b. The following definition of equivalence by finite decomposition is given: Two sets of points, A and B are equivalent by finite decomposition, Af=B, provided sets A1 , A2, ..., An and B1, B2, ..., Bn exist with the following properties:
(1) A=A1+A2+...+An B=B1+B2+...+Bn
(2) Aj • Ak=Bj • Bk=0 1 ≤ j < k ≤ n
(3) Aj≅Bj 1 ≤ j ≤ n
An historic measure problem is discussed briefly.
Two paradoxes of the sphere, the Hausdorff Paradox and the Banach and Tarski Paradox are stated and discussed in detail. The Hausdorff Paradox reads as follows: The surface K of the sphere can be decomposed into four disjoint subsets A, B, C, and Q such that (1) K=A+B+C+Q and (2) A≅B≅C, A≅B+C where Q is denumerable. A refinement of this Paradox is introduced in which the denumerable set Q is eliminated.
The Banach and Tarski Paradox states that in any Euclidean space of dimension n≥3, two arbitrary sets, bounded and containing interior points, are equivalent by finite decomposition. Various refinements of this paradox are noted. It is observed that the proofs of both paradoxes require the aid of the Axiom of Choice.
The controversy over the Axiom of Choice is discussed at length. A wide range of viewpoints is studied, ranging from total rejection by the intuitionists to practically complete acceptance of the axiom.
Seven theorems on cardinal numbers that are equivalent to the Axiom of Choice are listed. Six examples of theorems which require the aid of the Axiom of Choice in their proof are given.
Based on the results of Hausdorff, Banach and Tarski, and Robinson, three specific questions are answered as follows : with the aid of the Axiom of Choice (1) the surface of a sphere can be decomposed into subsets in such a way that a half and a third of the surface may be congruent to each other. (2) A solid sphere of fixed radius can be decomposed into a finite number of pieces and these pieces can be reassembled to form two solid spheres of the given radius. (3) The minimum number of pieces required in the above problem is five.
It is concluded that the general question, "Should the Axiom of Choice be accepted or rejected" is unanswerable at the present time. It is pointed out that the problem of existence and t he paradoxes that result from the axiom are major arguments against its use. However, the axiom simplifies many parts of set theory, analysis, and topology. The fact that Godel has proved the Axiom of Choice consistent with other generally accepted axioms of set theory, provided they are consistent with one another, is a second major point in its favor.
Finally, Appendix I contains some statements equivalent to the Axiom of Choice, and Appendix II contains some importru1t theoren1s of Banach and Tarski
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