Rather, we say that a real a is definable in the language of arithmetic ( or arithmetical ) if its Dedekind cut can be defined as a predicate in that language ; that is, if there is a first-order formula φ in the language of arithmetic, with two free variables, such that

Rather than there being any convenient date for the " fall of the Roman Empire " there was a progressive " de-Romanization " of the Western Roman Empire in Hispania and a weakening of central authority, throughout the 3rd, 4th and 5th centuries.

Rather, there are interpretations through which we understand the world and give it meaning.

Rather, the supposition that there are such things is the source of many philosophical confusions.

Rather than forcing programmers to adopt a particular style of programming, it permits several styles: object-oriented programming and structured programming are fully supported, and there are a number of language features which support functional programming and aspect-oriented programming ( including by metaprogramming and by magic methods ).

Rather than having a single production arm, there are a number of public broadcasting organizations that create programming for each of the three stations, each working relatively independently.

Dr. Kellner stated " Rather than cause brain damage, there is evidence that ECT may reverse some of the damaging effects of serious psychiatric illness.

Rather, God planned that there would be an earthquake ( or some such other natural disaster ) at that place and time, so that the city would fall to the Israelites.

Rather, Reform Jews focus on a future age-the World to Come-in which there is a perfected world of justice and mercy.

Rather than making extensive preparations for battlefield nuclear combat in Central Europe, the Soviet General Staff emphasized conventional military operations and believing that they had an advantage there.

Rather than having it professionally recorded, the band built their own solar-powered studio in Sacramento over five years and chose to produce the album themselves there.

Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible.

Rather than having a skilled worker measure every dimension of each part of the product against the plans or the other parts as it is being formed, there were jigs ready at hand to ensure that the part was made to fit this set-up.

Rather than order a new lens, the huge optic construction — twelve feet tall and with over a thousand prisms — was used there.

Rather, the being is brought here -- as is the case with all the other realms in the Buddhist cosmology -- by natural law: the law of karma, and they remain until the negative karma that brought them there has been used up.

In his autobiography, Rather notes that back then, television stations did not have their own radar systems, and there was no modern computerized radar that combines the radar image with an outline map.

Success there ( and a threat to bolt to ABC News ) helped Rather pull ahead of longtime correspondent Roger Mudd in line to succeed Walter Cronkite as anchor and Managing Editor of CBS Evening News.

Rather, it means that even the good which a person may intend is faulty in its premise, false in its motive, and weak in its implementation ; and there is no mere refinement of natural capacities that can correct this condition.

... Rather than deploy additional forces to Iraq, we believe the way forward is to begin the phased redeployment of our forces in the next four to six months, while shifting the principal mission of our forces there from combat to training, logistics, force protection and counter-terror.

Rather than just using a thick tube, there are similar techniques used to the same effect such as replacing thin wire elements with cages to simulate a thicker element.

Rather, there is a vetting process of varying sorts.

Rather than the subject being one-sided or biased towards one view-point, there is a strong balance of opinions in the film, with some interviewees who are happy with their living situation, some who are not, and some who have a neutral opinion.

Rather than one level of subsistence, there are numerous levels at which a population and a food supply can be in equilibrium in the sense that they can be indefinitely sustained.

" Rather, " God's kindness is meant to lead you to repentance " ( Romans 2: 4, ESV ); nonetheless, there has traditionally been a stress on reconciliation as a precondition to fellowship.

Rather what Kornbluth and Pohl are really doing is warning against the dangers inherent in perfecting `` a science of man and his motives ''.

Rather, such assignments are made, as they must be, on the basis of certain overall rules and standards, representing to some extent a statistical approach to the problem, taking into account for each situation some of the variables ( e.g., power and station separations ) and averaging out others in order to achieve the balance which must be struck between protection against destructive interference and the assignment of a number of stations large enough to afford optimum radio service to the Nation.

Rather, meat and potatoes, sugar, and dairy products are the main sources of carbohydrate, protein, oils, and fats.

Rather, they are impressed with the British Government's success in forcing -- and helping -- the British textile industry to shrink and to change over to other products.

Rather they are genuine clods, proud of their cloddishness and openly antagonistic to the illuminating influences of aesthetics or thought.

Rather, the sample is evaporated in a controlled manner ( field evaporation ) and the evaporated ions are impacted onto a detector, which may be up to several meters from the specimen.

Rather, given the implications of the theory of oral-formulaic composition and oral tradition, the question concerns how the poem is to be understood, and what sorts of interpretations are legitimate.

Rather than being scrapped or converted for other uses, sometimes retired buses are saved for preservation.

Rather, in modern times, the various autonomous houses have formed themselves loosely into congregations ( for example, Cassinese, English, Solesmes, Subiaco, Camaldolese, Sylvestrines ) that in turn are represented in the Benedictine Confederation that came into existence through Pope Leo XIII's Apostolic Brief " Summum semper " on July 12, 1883.

Rather than being written as history, the Deuteronomistic history – Deuteronomy, Joshua, Judges, Samuel and Kings – was intended to illustrate a theological scheme in which Israel and her leaders are judged by their obedience to the teachings and laws ( the covenant ) set down in the book of Deuteronomy.

Rather, they are acts of hesed that go beyond measure and demonstrate that a person can go beyond the minimum expectations of the law and choose the unexpected.

Rather than being ontologically real, in Christian Science evil and its manifestations are instead terrible lies about God and His creation.

Rather, the independent congregations are a network with each congregation participating at its own discretion in various means of service and fellowship with other congregations ( see Sponsoring church ( Churches of Christ )).

Rather, carinates are unique in having, for example, a globe-shaped, convex head on the humerus and fully fused bones in the lower leg and outer arm.

Rather, the system of assignment of sound values is much more systematic in the Cirth than in the historical runes ( e. g., voiced variants of a voiceless sound are expressed by an additional stroke ).

Rather, the items are grouped together because of a family resemblance.

Rather, evil spirits described in various faiths ' traditions, such as Satan, fallen angels, demons, and jinns, are metaphors for the base character traits a human being may acquire and manifest when he turns away from God and follows his lower nature.

Rather, he maintained, our beliefs are more a result of accumulated habits, developed in response to accumulated sense experiences.

Rather the form of the argument is generalized to considering configurations, which are connected subgraphs of G with the degree of each vertex ( in G ) specified.

Rather they are charges that are easy for Loki to make and difficult for his targets to disprove, or which they do not care to refute.

Rather, they are to be understood as eight significant dimensions of one's behaviour — mental, spoken, and bodily — that operate in dependence on one another ; taken together, they define a complete path, or way of living.

Rather than harmonics which are exact multiples of the fundamental, as generated by organ pipes, the Hammond organ uses the nearest-available frequencies generated by tonewheels, a factor which has some part in creating its distinctive tone color.

Rather than purchasing a special purpose rifle, which many novice hunters would outgrow within a few hunting seasons, a single rifle can be used with special handloaded rounds until such time more powerful rounds are desired and become appropriate.

Rather uncommon scales such as whole tone scale are also typical for this movement.