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).

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 The church of Santa Comba de Bande and early medieval Iberian architecture: new chronological results

Antiquity ◽  
2017 ◽  
Vol 91 (358) ◽  
pp. 1011-1026 ◽  
Author(s):  
José C. Sánchez-Pardo ◽  
Rebeca Blanco-Rotea ◽  
Jorge Sanjurjo-Sánchez
Keyword(s):  
Early Medieval ◽  
Image Position ◽  
The Church

Abstract

scholarly journals Coq without Type Casts: A Complete Proof of Coq Modulo Theory

10.29007/bjpg ◽  
2018 ◽  
Author(s):  
Jean-Pierre Jouannaud ◽  
Pierre-Yves Strub
Keyword(s):  
Theoretical Study ◽  
Type System ◽  
Order Theory ◽  
Formal Proof ◽  
Complete Proof ◽  
Type Checking ◽  
First Order ◽  
First Order Theory ◽  
Good Trade ◽  
The Church

Incorporating extensional equality into a dependent intensional type system such as the Calculus of Constructions provides with stronger type-checking capabilities and makes the proof development closer to intuition. Since strong forms of extensionality lead to undecidable type-checking, a good trade-off is to extend intensional equality with a decidable first-order theory T, as done in CoqMT, which uses matching modulo T for the weak and strong elimination rules, we call these rules T-elimination. So far, type-checking in CoqMT is known to be decidable in presence of a cumulative hierarchy of universes and weak T-elimination. Further, it has been shown by Wang with a formal proof in Coq that consistency is preserved in presence of weak and strong elimination rules, which actually implies consistency in presence of weak and strong T-elimination rules since T is already present in the conversion rule of the calculus.We justify here CoqMT’s type-checking algorithm by showing strong normalization as well as the Church-Rosser property of β-reductions augmented with CoqMT’s weak and strong T -elimination rules. This therefore concludes successfully the meta-theoretical study of CoqMT.

A Problem of Herstein on Group Rings

1974 ◽  
Vol 17 (2) ◽  
pp. 201-202 ◽  
Author(s):  
Edward Formanek
Keyword(s):  
Group Ring ◽  
Finite Subset ◽  
Zero Divisor ◽  
Formal Proof ◽  
Left Zero ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position ◽  
Finite Sum

Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let gl,…,gn be a finite subset of G, and let1—x is not a zero divisor so it is invertible and its inverse is 1+x+x3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].

Spiritual Healing, Sick Kids and the Law: Inequities in the American Healthcare System

2003 ◽  
Vol 29 (2-3) ◽  
pp. 269-299
Author(s):  
Janna C. Merrick
Keyword(s):  
Medical Science ◽  
Spiritual Healing ◽  
Best Interests ◽  
Legal Rights ◽  
High Tech ◽  
Divine Love ◽  
Religious Minority ◽  
Human Need ◽  
East End ◽  
The Church

Main Street in Sarasota, Florida. A high-tech medical arts building rises from the east end, the county's historic three-story courthouse is two blocks to the west and sandwiched in between is the First Church of Christ, Scientist. A verse inscribed on the wall behind the pulpit of the church reads: “Divine Love Always Has Met and Always Will Meet Every Human Need.” This is the church where William and Christine Hermanson worshipped. It is just a few steps away from the courthouse where they were convicted of child abuse and third-degree murder for failing to provide conventional medical care for their seven-year-old daughter.This Article is about the intersection of “divine love” and “the best interests of the child.” It is about a pluralistic society where the dominant culture reveres medical science, but where a religious minority shuns and perhaps fears that same medical science. It is also about the struggle among different religious interests to define the legal rights of the citizenry.

Ideas and Leaders in Contentious Politics: One Parish Priest in El Salvador's Popular Movement

2014 ◽  
Vol 38 (01) ◽  
pp. 76-101
Author(s):  
PETER M. SANCHEZ
Keyword(s):  
Collective Action ◽  
El Salvador ◽  
Liberation Theology ◽  
Contentious Politics ◽  
Religious Leaders ◽  
Political Movement ◽  
Parish Priest ◽  
Historical Moment ◽  
Resource Mobilisation ◽  
The Church

AbstractThis paper examines the actions of one Salvadorean priest – Padre David Rodríguez – in one parish – Tecoluca – to underscore the importance of religious leadership in the rise of El Salvador's contentious political movement that began in the early 1970s, when the guerrilla organisations were only just beginning to develop. Catholic leaders became engaged in promoting contentious politics, however, only after the Church had experienced an ideological conversion, commonly referred to as liberation theology. A focus on one priest, in one parish, allows for generalisation, since scores of priests, nuns and lay workers in El Salvador followed the same injustice frame and tactics that generated extensive political mobilisation throughout the country. While structural conditions, collective action and resource mobilisation are undoubtedly necessary, the case of religious leaders in El Salvador suggests that ideas and leadership are of vital importance for the rise of contentious politics at a particular historical moment.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

Sign in / Sign up

Export Citation Format

Share Document