With the resumption of Soviet testing and their intransigence at the Geneva talks, however, the hope that this third choice would prove viable has been shaken.

The second choice, full testing, has become even more risky just because the current Soviet tests have already dangerously contaminated the atmosphere.

The problem arises, if it does arise, when the educator has to make a choice or a decision within the area of his professional competence, but which bears some relation to the social structure.

`` There had been a threesome at the party in the suite's bedroom: Miss Harrington ( this was Diane's choice for a Roman name ), another woman who has figured in other very interesting events and one of your well-known American actors.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

: For any set A, the power set of A ( with the empty set removed ) has a choice function.

: Every set has a choice function.

The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.

In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element ; this holds trivially.

( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

This has been used as an argument against the use of the axiom of choice.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

Science fiction set in what was the future but is now the past, like Arthur C. Clarke's 2001: A Space Odyssey or Nineteen Eighty-Four, are not alternate history because the author has not made the conscious choice to change the past.

* Man has free will to respond or resist: Free will is limited by God's sovereignty, but God's sovereignty allows all men the choice to accept the Gospel of Jesus through faith, simultaneously allowing all men to resist.

The power of choice under this definition has the potential for as much harm as it does good, and open theists see free will as the best answer to the problem of evil.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

What it is trying to do is to protect the little man, too, as well as trying to maintain a flow of fresh meat to all stores, with choice of cut being made by the consumer, not the store.

Johnston, who had little choice in allowing Floyd and Pillow to take charge at Fort Donelson on the basis of seniority after he ordered them to add their forces to the garrison, took the blame and suffered calls for his removal because a full explanation to the press and public would have exposed the weakness of the Confederate position.

The choice of isotope ordinarily has little effect on the electrical properties of a material, but does affect the frequency of lattice vibrations, this effect suggested that superconductivity be related to vibrations of the lattice.

Like Case, many cyberpunk protagonists are manipulated, placed in situations where they have little or no choice, and although they might see things through, they do not necessarily come out any further ahead than they previously were.

The guest would be offered the meal table d ' hôte, which is a meal offered at a set price with very little choice of dishes, sometimes none at all.

Currie had little experience in the politics of the Republic and was widely seen as the party's last choice, nominated only when no-one else was available.

Gregory VI, seeing that little choice was left to him, complied of his own accord and laid down his office.

" School choice " as it entails a switch from public to private schooling would therefore do little to solve the problems facing the educational system, since a private school would perform no better than a public school when faced with exactly the same student body.

Austin saw little choice but revolution.

The experience of the failure to contain the British attacks at Ypres and the drastic reduction in areas of the western front which could be considered ' quiet ', after the tank and artillery surprise at Cambrai, left the German Oberste Heeresleitung with little choice but to return to a strategy of decisive victory in 1918.

The researchers concluded that " the change happened gradually, in a way consistent with evolution via natural selection — not suddenly, as researchers once had little choice but to believe.

Hunold, who originally resisted acknowledging Charles as overlord, soon had little choice.

Ecological selection can be said to be taking place in any circumstance where inheritance of specific traits is determined by ecology alone without direct sexual competition, when e. g. sexual competition is strictly ecological or economic, there is little or no mate choice, females do not resist any male who wishes to mate, all traits will be equally propagated regardless of mating, or the species is hermaphroditic or asexually reproducing, an ecological selection is taking place.

Walpole commanded a substantial majority in Parliament and George II had little choice but to accept him or risk ministerial instability.

Barriers to exit increase the intensity of competition in an industry because existing firms have little choice but to stay and fight when market conditions have deteriorated.

Often wrestlers have little choice in their name, which is given to them by their trainer ( or stablemaster ), or by a supporter or family member who encouraged them into the sport.

At the moment of electoral victory, Bonforte dies of the aftereffects of his kidnapping, and Smythe realizes he has little choice but to assume the role for life.

To mount a massive invasion of Europe from England, military planners had little choice but to stage units around the country with those that would land first nearest to the embarkation point.

Takahashi determined that the peacock ’ s train was not the universal target of female mate choice, showed little variance across male populations, and, based on physiological data collected from this group of peafowl, do not correlate to male physical conditions.

Its cultural impact was therefore significant since many Canadians had little or no choice for their information and entertainment other than from these two powerful media.

John had little choice but to leave Syria with his ambitions only partially realised.

Traveling through many public places such as government offices is hardly optional for most people, yet consumers have little choice but to submit to companies ' surveillance practices.

His opponents, being suddenly confronted with a new threat and with little reserves, had no choice but to weaken the area closest to the flanking formation and draw up a battle line at a right angle in an attempt to stop this new threat.

Faced with hundreds of thousands of dollars in construction costs to wind down a project that could not garner funding, McGill was left with little choice but to trade Bull for title to the Highwater equipment.