These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

( Here " almost all " has the sense " all but a countable set "; see Properties below.

In this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable.

In terms of ultra vires actions in the broad sense, a reviewing court may set aside an administrative decision if it is unreasonable ( under Canadian law, following the rejection of the " Patently Unreasonable " standard by the Supreme Court in Dunsmuir v. New Brunswick ), Wednesbury unreasonable ( under British law ), or arbitrary and capricious ( under U. S. Administrative Procedure Act and New York State law ).

The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set.

The term epískopos was not from the earliest times clearly distinguished from the term presbýteros (" elder ", " senior ", nowadays used to signify a priest ), but the term was already clearly used in the sense of the order or office of bishop, distinct from that of priest in the writings of Ignatius of Antioch ( died c. 108 ), and sources from the middle of the 2nd century undoubtedly set forth that all the chief centres of Christianity recognized and had the office of bishop, using a form of organization that remained universal until the Protestant Reformation.

However, the term naive set theory is also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory ; care is required to tell which sense is intended.

If there is a sense in which a card game can have an " official " set of rules, it is when that card game has an " official " governing body.

They are the only tiles in the whole set that don't match other tiles in the normal sense.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness — originally called bicompactness — involves families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

In other words, the enumerable first n bits of Omega are highly compressible in the sense that they are limit-computable by a very short algorithm ; they are not random with respect to the set of enumerating algorithms.

( The system is not limited to alphabets in the strict technical sense ; languages that use a syllabary or abugida, for example Cherokee, can use the same ordering principle provided there is a set ordering for the symbols used.

In 1966, several rock releases were arguably concept albums in the sense that they presented a set of thematically-linked songs-and they also instigated other rock artists to consider using the album format in a similar fashion: The Beach Boys ' Pet Sounds was a musical portrayal of Brian Wilson's state of mind at the time ( and a major inspiration to Paul McCartney ).

Nevertheless, if a function satisfies the Cauchy – Riemann equations in an open set in a weak sense, then the function is analytic.

Divisio is an exhaustive list of subsets of a set, in the sense that every member of the " divided " set is a member of one of the subsets.

Though the resolution is historically significant, Hideaki Uemura, professor at Keisen University in Tokyo and a specialist in indigenous peoples ' rights, commented that the motion is " weak in the sense of recognizing historical facts " as the Ainu were " forced " to become Japanese in the first place.

The first nineteen articles of the “ Grundgesetz ” represent the constitution in narrower sense and determine the fundamental rights of German citizens.

Citizenship became an idealized, almost abstract, concept, and did not signify a submissive relation with a lord or count, but rather indicated the bond between a person and the state in the rather abstract sense of having rights and duties.

In this sense, citizenship was described as " a bundle of rights -- primarily, political participation in the life of the community, the right to vote, and the right to receive certain protection from the community, as well as obligations.

Even though the rights of the individual were not secured by the Athenian constitution in the modern sense ( the ancient Greeks had no word for " rights "), the Athenians enjoyed their liberties not in opposition to the government but by living in a city that was not subject to another power and by not being subjects themselves to the rule of another person.

The modern sense of human rights can be traced to Renaissance Europe and the Protestant Reformation, alongside the disappearance of the feudal authoritarianism and religious conservativism that dominated the Middle Ages.

That this Assembly doth explicitly and peremptorily declare, that it views the powers of the federal government as resulting from the compact to which the states are parties, as limited by the plain sense and intention of the instrument constituting that compact, as no further valid than they are authorized by the grants enumerated in that compact ; and that, in case of a deliberate, palpable, and dangerous exercise of other powers, not granted by the said compact, the states, who are parties thereto, have the right, and are in duty bound, to interpose, for arresting the progress of the evil, and for maintaining, within their respective limits, the authorities, rights and liberties, appertaining to them.

Objectivism's central tenets are that reality exists independent of consciousness, that human beings have direct contact with reality through sense perception, that one can attain objective knowledge from perception through the process of concept formation and inductive logic, that the proper moral purpose of one's life is the pursuit of one's own happiness ( or rational self-interest ), that the only social system consistent with this morality is full respect for individual rights embodied in laissez-faire capitalism, and that the role of art in human life is to transform humans ' metaphysical ideas by selective reproduction of reality into a physical form — a work of art — that one can comprehend and to which one can respond emotionally.

Most broadly and concisely, property in the legal sense refers to the rights of people in or over certain objects or things.

Squatters often claim rights over the spaces they have squatted by virtue of occupation, rather than ownership ; in this sense, squatting is similar to ( and potentially a necessary condition of ) adverse possession, by which a possessor of real property without title may eventually gain legal title to the real property.

Act utilitarians, on the other hand, do not accept human rights as moral principles in and of themselves, but that does not mean that they reject them altogether: first, most act utilitarians, as explained above, would agree that acts such as enslavement and genocide always cause great unhappiness and very little happiness ; second, human rights could be considered rules of thumb so that, although torture might be acceptable under some circumstances, as a rule it is immoral ; and, finally, act utilitarians often support human rights in a legal sense because utilitarians support laws that cause more good than harm.

* Natural rights are rights which are " natural " in the sense of " not artificial, not man-made ", as in rights deriving from deontic logic, from human nature, or from the edicts of a god.

Some thinkers see rights in only one sense while others accept that both senses have a measure of validity.

In one sense, a right is a permission to do something or an entitlement to a specific service or treatment, and these rights have been called positive rights.

However, in another sense, rights may allow or require inaction, and these are called negative rights ; they permit or require doing nothing.

The general concept of rights is that they are possessed by individuals in the sense that they are permissions and entitlements to do things which other persons, or which governments or authorities, can not infringe.

But there is another sense of group rights in which people who are members of a group can be thought of as having specific individual rights because of their membership in a group.

Neither is primary experience understood according to the attitude of modern empiricism in which nothing is thought to be received other than signals of sensory qualities producing their responses in the appropriate sense organs.

These desires presuppose a sense of causally efficacious powers in which one is involved, some working for one's good, others threatening ill.

he is questioning, also, every epistemology which stems from Hume's presupposition that experience is merely sense data in abstraction from causal efficacy, and that causal efficacy is something intellectually imputed to the world, not directly perceived.

And we had the uneasy sense that the cleavage between the moral and the political progressed amid the events which concern us.

Neither the vibrant enthusiasm which bespeaks a people's intuitive sense of the fitness of things at climactic moments nor the vital argumentation betraying its sense that something significant has transpired was in evidence.

He has employed from his section rich immediate materials which in a loose sense can be termed Southern.

In a sense, Einstein's theory is simpler than Newton's, and there is a corresponding sense in which Copernicus' theory is simpler than Ptolemy's.

I fled, however, not from what might have been the natural fear of being unable to disguise from you that the things about my bridegroom -- in the sense you meant the word `` things '' -- which you had been galvanizing yourself to tell me as a painful part of your maternal duty were things which I had already insisted upon finding out for myself ( despite, I may now say, the unspeakable awkwardness of making the discovery on principle, yes, on principle, and in cold blood ) because I was resolved, as a modern woman, not to be a mollycoddle waiting for Life but to seize Life by the throat.

The only rules which I think we shall follow will be those of common sense, justice, and fairness ''.

This, no doubt, is part of what Gilbert Seldes implies when he says of the arts, `` They give form and meaning to life which might otherwise seem shapeless and without sense ''.

The terms `` renewal '' and `` refreshed '', which often come up in aesthetic discussion, seem partly to derive their import from the `` renewal '' of purpose and a `` refreshed '' sense of significance a person may receive from poetry, drama, and fiction.

In any inquiry into the way in which great literature affects the emotions, particularly with respect to the sense of harmony, or relief of tension, or sense of `` a transformed inner nature '' which may occur, a most careful exploration of the particular feature of the experience which produces the effect would be required.

In the calm which follows the reading of a poem, for example, is the effect produced by the enforced quiet, by the musical quality of words and rhythm, by the sentiments or sense of the poem, by the associations with earlier readings, if it is familiar, by the boost to the self-esteem for the semi-literate, by the diversion of attention, by the sense of security in a legitimized withdrawal, by a kind license for some variety of fantasy life regarded as forbidden, or by half-conscious ideas about the magical power of words??

His father was a professor at Hartford Theological Seminary, and from him he acquired a conviction, which he passed along to me, that there is in the universe of persons a moral law, the law of love, which is a natural law in the same sense as is the physical law.

Each will decide on his own course somewhere between these two extreme cases according to the sense of responsibility which is determined for him by the particular circumstances of his own life.