A formal proof of Gödel's theorem

1939 ◽  
Vol 4 (2) ◽  
pp. 61-68
Author(s):  
L. Chwistek
Keyword(s):  
Recent Result ◽  
Formal Proof ◽  
Principal Term ◽  
Elementary System ◽  
Ancestral Function ◽  
Image Position ◽  
Simple Systems ◽  
System 1 ◽  
Gödel’S Theorem

I. The present paper contains a formal proof of the following theorem of the elementary system ⊨ [ 1 0 ] c:This theorem means that if there were a theorem of ⊢0 ( 3 2 ) 0 stating that there is no contradiction in ⊢2 ( 5 4 ) 2, we would have a contradiction in ⊢2 ( 5 4 ) 2. (This is in accordance with the second theorem of Gödel.)Our proof is based on the calculus of the system ⊨ [ 1 0 ] c and of the metasystem ⊢0 ( 3 2 ) 0.Note that we could takeinstead of Ax E .1 ( L ) (.1 ( L ) .0 ( L ) L) Z. We would then have to do with the simple systems and metasystems of Hetper, which do not contain propositional variables or the logico-semantical axiom. Our method applies equally to the systems and metasystems of New Foundation and to the simple systems and metasystems of Hetper.Our proof can be considerably simplified by using a recent result of Hetper concerning ancestral functions. Hetper introduces the following abbreviations:If A ( Z1c), B ( x1cy1cZ1c) are propositions, the expression will be called an ancestral function (we prove without difficulty that this expression is a proposition). The expressions will be called respectively the principal term and the term of derivation of this function.

scholarly journals ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k3− 4k}

2007 ◽  
Vol 49 (2) ◽  
pp. 333-344 ◽  
Author(s):  
YANN BUGEAUD ◽  
ANDREJ DUJELLA ◽  
MAURICE MIGNOTTE
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
The Family ◽  
Diophantine Triples ◽  
Image Position ◽  
Diophantine Quadruples

AbstractIt is proven that ifk≥ 2 is an integer anddis a positive integer such that the product of any two distinct elements of the setincreased by 1 is a perfect square, thend= 4kord= 64k5−48k3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k− 1,k+ 1,c,d} are regular.

A formal theorem in Church's theory of types

1942 ◽  
Vol 7 (1) ◽  
pp. 28-33 ◽  
Author(s):  
M. H. A. Newman ◽  
A. M. Turing
Keyword(s):  
Formal Proof ◽  
Image Position ◽  
The Church ◽  
Logical Formalism

This note is concerned with the logical formalism with types recently introduced by Church [1] (and called (C) in this note) It was shewn in his paper (Theorem 26α) that if Yα stands for(a form of the “axiom of infinity” for the type α), Yα can be proved formally, from Yι and the axioms 1 to 7, for all types α of the forms ι′, ι″, …. For other types the question was left open, but for the purposes of an intrinsic characterisation of the Church type-stratification given by one of us, it is desirable to have the remaining cases cleared up. A formal proof of Yα is now given for all types α containing ι, but the proof uses, in addition to Axioms 1 to 7 and Yι, also Axiom 9 (in connection with Def. 4), and Axiom 10 (in Theorem 9).

On the global attractivity in a generalized delay-logistic differential equation

1986 ◽  
Vol 100 (1) ◽  
pp. 183-192 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Differential Equation ◽  
Trivial Solution ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Countable Number ◽  
Local Asymptotic Stability ◽  
Discrete Delays ◽  
Image Position ◽  
System 1

The purpose of this article is to derive a set of ‘easily verifiable’ sufficient conditions for the local asymptotic stability of the trivial solution ofand then examine the ‘size’ of the domain of attraction of the trivial solution of the nonlinear system (1·1) with a countable number of discrete delays.

scholarly journals Introduction to Diophantine Approximation. Part II

2017 ◽  
Vol 25 (4) ◽  
pp. 283-288
Author(s):  
Yasushige Watase
Keyword(s):  
Diophantine Approximation ◽  
Irrational Number ◽  
Basic Result ◽  
Formal Proof ◽  
Integer Solution ◽  
Linear Forms ◽  
Diophantine Approximations ◽  
Integer Solutions ◽  
System 1 ◽  
Hurwitz Theorem

SummaryIn the article we present in the Mizar system [1], [2] the formalized proofs for Hurwitz’ theorem [4, 1891] and Minkowski’s theorem [5]. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in [3], [6]. A formal proof of Dirichlet’s theorem, namely an inequation |θ−y/x| ≤ 1/x2has infinitely many integer solutions (x, y) where θ is an irrational number, was given in [8]. A finer approximation is given by Hurwitz’ theorem: |θ− y/x|≤ 1/√5x2. Minkowski’s theorem concerns an inequation of a product of non-homogeneous binary linear forms such that |a1x + b1y + c1| · |a2x + b2y + c2| ≤ ∆/4 where ∆ = |a1b2− a2b1| ≠ 0, has at least one integer solution.

Solitary wave solutions for a model of the two-way propagation of water waves in a channel

1981 ◽  
Vol 90 (2) ◽  
pp. 343-360 ◽  
Author(s):  
J. F. Toland
Keyword(s):  
Initial Data ◽  
Water Waves ◽  
Coupled System ◽  
Finite Amplitude ◽  
Solitary Wave Solutions ◽  
Flat Bottom ◽  
One Dimensional ◽  
Image Position ◽  
System 1 ◽  
Relevant Class

Bona and Smith (6) have suggested that the coupled system of equationshas the same formal justification as other Boussinesq-type models for the two-way propagation of one-dimensional water waves of small but finite amplitude in a channel with a flat bottom. The variables u and η represent the velocity and elevation of the free surface, respectively. Using the energy invariantthey show that for a restricted, but nevertheless physically relevant, class of initial data, the system (1·1) has solutions which exist for all time, and that in such circumstances the wave height is bounded solely in terms of the initial data.

On the iteration of a continuous mapping of a compact space into itself

1950 ◽  
Vol 46 (1) ◽  
pp. 46-56 ◽  
Author(s):  
F. G. Friedlander
Keyword(s):  
Continuous Mapping ◽  
Transformation Theory ◽  
Dependent Variables ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Mechanics ◽  
Image Position ◽  
System 1 ◽  
Lipschitz Conditions ◽  
Elementary Topology

1. The questions considered in this note are suggested by the elementary topology of the trajectories of systems of non-linear differential equations. Such a system may be assumed in the formand the values of the dependent variables x1, x2, …, xn at ‘time’ t can be represented by a point P(t) in a ‘phase space’ . As t varies, P(t) describes a curve in , which is a trajectory of (1). Now it often happens that contains a subspace E (usually of lower dimension) with the following properties: (i) by considering the trajectories generated by points P(t) which are, for t = 0, in E, all the trajectories of (1) are obtained; (ii) if P(0) is in E, then P(t) is not in E for 0 < t < c, where c is a constant independent of P(0) in E; (iii) if P(0) is in E, then the trajectory meets E again for some finite t at a point P(T) (T is not necessarily the same for all points of E). By considering P(T) as the image of P(O), a mapping of E into itself is defined which is associated with the system (1), and the topology of the trajectories of (1) can be studied conveniently by discussing this mapping. When the functions fi in (1) satisfy the continuity and Lipschitz conditions of the classical existence-and-uniqueness theorem, the mapping is one-one and continuous. The study of this ‘transformation theory’, initiated by Poincaré, has been developed chiefly by G. D. Birkhoff(l,2). His results have been applied to problems of ‘non-linear mechanics’ by N. Levinson(3).

A Note on Upper Bounds for the Eigenvalues ofy″ + λpy=0

1975 ◽  
Vol 18 (3) ◽  
pp. 347-351
Author(s):  
Rodney D. Gentry
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Mass Distribution ◽  
Unit Length ◽  
Upper Bounds ◽  
Natural Modes ◽  
Negative Function ◽  
Modes Of Vibration ◽  
Image Position ◽  
System 1

The natural modes of a small planar transversal vibration of a fixed string of unit length and tension are determined by the eigenvalues and associated eigenfunctions of the differential equation(1)subject to the boundary condition(2)where the non-negative functionpdescribes the mass distribution of the string. That the distribution of mass on the string influences the modes of vibration, may be reflected by observing that the eigenvalues determined by the system (1–2) may be considered functions of the densityp, λn(p), where λ1(p)<λ2(p)<….

scholarly journals Schmidt’s game, fractals, and orbits of toral endomorphisms

2010 ◽  
Vol 31 (4) ◽  
pp. 1095-1107 ◽  
Author(s):  
RYAN BRODERICK ◽  
LIOR FISHMAN ◽  
DMITRY KLEINBOCK
Keyword(s):  
Hausdorff Dimension ◽  
Recent Result ◽  
Lacunary Sequence ◽  
Alternative Proof ◽  
Integer Matrix ◽  
Image Position ◽  
Schmidt’S Game ◽  
Badly Approximable

AbstractGiven an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

Cubic systems with four real line invariants

1995 ◽  
Vol 118 (1) ◽  
pp. 7-19 ◽  
Author(s):  
Robert E. Kooij
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Quadratic Nonlinearities ◽  
Mathematical Problems ◽  
Image Position ◽  
Open Question ◽  
System 1 ◽  
Cubic Systems

A polynomial system is a real autonomous system of ordinary differential equations on the plane with polynomial nonlinearities:with aij, bij ∈ ℝ and where x = x(t) and y = y(t) are real-valued functions.The problem of analysing limit cycles (isolated periodic solutions) in polynomial systems was first discussed by Poincaré[16]. Then, in the famous list of 23 mathematical problems stated in 1900, Hilbert[9] asked in the second part of the 16th problem for an upper bound for the number of limit cycles for nth degree polynomial systems, in terms of n. Recently, it has been proved that, given a particular choice of coefficients for a system of form (1·1), the number of limit cycles is finite. This result is known as Dulac's theorem, see Ecalle[8] or Il'yashenko[10]. However, it is unknown whether or not there exists an upper bound for the number of limit cycles in system (1·1) in terms of n. Even for quadratic systems (i.e. polynomial systems with quadratic nonlinearities) this remains an open question.

scholarly journals Intersection theorems for systems of sets (III)

1974 ◽  
Vol 18 (1) ◽  
pp. 22-40 ◽  
Author(s):  
P. Erdös ◽  
E. C. Milner ◽  
R. Rado
Keyword(s):  
Index Set ◽  
System A ◽  
Image Position ◽  
System 1

A system or family (Aγ: γ∈ N) of sets Aγ, indexed by the elements of a set N, is called an (a, b)-system if ¦N¦ = a and ¦Aγ¦ = b for γ ∈ N. Expressions such as “(a, <b)-system” are self-explanatory. The system (Aγ: γ∈N) is called a δ-system [1] if Aμ∩Aγ = Ap ∩ Aσ whenever μ, γ, ρ, σ ∈ N; μ≠ γ; ρ ≠ σ. If we want to indicate the cardinality ¦N¦ of the index set N then we speak of a δ(¦N¦) system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a δ(c)-subsystem. In [2], for every choice of cardinals b, c such that the least cardinal a = fδ(b, c) was determined which has the property that every (a, < b)-system contains a δ(c)-subsystem.

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