The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.

The analgesic choice is also determined by the type of pain: for neuropathic pain, traditional analgesics are less effective, and there is often benefit from classes of drugs that are not normally considered analgesics, such as tricyclic antidepressants and anticonvulsants.

As a type of face-to-face communication, body language and choice tonality play a significant role, and may have a greater impact upon the listener than informational content.

" One commentary characterizes the remark as making economics an approach rather than a subject matter but with great specificity as to the " choice process and the type of social interaction that analysis involves.

This type of argument can occur in policy debates regarding regulation of industries, as well as in relation to personal choice of lifestyle.

This rather strange data type is perfectly workable, in some ways simply the opposite of the more conventional choice of representing only integers.

A false dilemma ( also called false dichotomy, the either-or fallacy, fallacy of false choice, black-and-white thinking, or the fallacy of exhaustive hypotheses ) is a type of logical fallacy that involves a situation in which only two alternatives are considered, when in fact there is at least one additional option.

In intuitionistic theories of type theory ( especially higher-type arithmetic ), many forms of the axiom of choice are permitted.

A pen plotter's speed is primarily limited by the type of pen used, so the choice of pen is a key factor in pen plotter output speed.

However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility.

Critics of this traditional view have observed that the types can be quite strongly stereotyped by professions ( although neither Myers nor Keirsey engaged in such stereotyping in their type descriptions ), and thus may arise more from the need to categorize people for purposes of guiding their career choice.

::# A point-to-point link ( described above ) is sometimes categorized as a special instance of the physical star topology – therefore, the simplest type of network that is based upon the physical star topology would consist of one node with a single point-to-point link to a second node, the choice of which node is the ' hub ' and which node is the ' spoke ' being arbitrary.

A more classical-liberal approach that dates from the 1970s that denotes ' public-choice ' theory type approaches which question the benevolence of social planners to maximize the utility of a representative individual and instead stress how political forces affect the choice of policies that may not be so benevolent.

The choice and sequence of rotation crops depends on the nature of the soil, the climate, and precipitation which together determine the type of plants that may be cultivated.

The latitude where the garlic is grown affects the choice of type as garlic can be day-length sensitive.

Given the type of warfare that had been carried out in the conflict — wherein highly mobile groups of soldiers and rebels continued to move to and from on the desert terrain, retreating at a time and then suddenly attacking to regain control of small towns and villages in the Eastern rebel held parts of Libya — had led to the technical becoming a vehicle of choice for both sides.

Advances in equipment and fabrication techniques give the sign or display manufacturer an extremely wide choice of production techniques, ranging from injection molding for intricate, smaller-size, mass-production signs ( generally 5000 units is the minimum ) to vacuum and pressure forming for larger signs of limited runs.

It was Britain's second best selling car ( behind the Ford Escort ) in 1984 and 1985, and at its peak, this version of the Cavalier came with the choice of 1. 3 or 1. 6 L engines derived from the smaller Vauxhall Astra ( also sold as the ( Opel Kadett ), while for 1983 a 1. 8 L engine was launched, which had electronic fuel injection.

If the person was sentenced before that date, they have the choice of the electric chair or lethal injection.

For person sentenced before 25 March 1994, the condemned is given the choice between the gas chamber and lethal injection.

The state offered inmates the choice of either lethal injection or hanging for their execution method, the latter being the default.

Also available were two PSA ( non-electronically controlled Lucas CAV injection pumps ) Indirect injection diesel engines, with the choice of naturally aspirated 1. 9-litre XUD9 or turbocharged 1. 8 XUD7T engines.

This would be used in midgets and sprints into the 1960s, with a choice of carburetor or Hilborn fuel injection.

It was launched with the choice of a 3. 0 liter V6 with direct injection derived from the 3. 6 liter unit in the Cadillac CTS, or a 2. 8 liter turbocharged V6.

The treatment of choice is a single dose of benzathine benzylpenicillin given by intramuscular injection, or a five-day to one-week course of either oral penicillin or intramuscular procaine benzylpenicillin.

Dependency injection is a software design pattern that allows a choice of component to be made at run-time rather than compile time.

When a patient displays a cardiac flatline, the treatment of choice is an injection of vasopressin ( epinephrine and atropine are also possibilities ) and chest compressions.

The choice of the heading technique is dependent upon the accuracy requirements, field conditions, and the time available to accomplish the heading.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

This technique of dyeing is unique, in that the final color is controlled by the choice of the diazoic and coupling components.

However, with the proper choice of h, one can use the technique of rolling hash to compute all those hashes with an effort proportional to k + n.

) In order to assess these structures radiographically, a technique called an intravenous pyelogram was historically utilized, and today at many institutions CT urography is the technique of choice.

Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis ( CH ), nor the axiom of choice, can be proved from the standard Zermelo – Fraenkel axioms ( ZF ) of set theory.

It can be the method of choice for catching the elusive muskie, and a useful technique for catching walleye, black bass and striped bass.

The choice of weapon and knowing the opponents ' weapons is detrimental to the assessment of the right technique and strategy.

The imaging technique of choice for cystic echinococcosis is ultrasonography since it is not only able to visualize the cysts in the body ’ s organs but it is also inexpensive, non-invasive and gives instant results.

As with cystic echinococcosis, ultrasonography is the imaging technique of choice for alveolar echinococcosis and is usually complemented by CT scans since CT scans are able to detect the largest number of lesions and calcifications that are characteristic of alveolar echinococcosis.

All this led to the choice of Code Excited Linear Prediction ( CELP ) as the encoding technique to use for Speex.

The ability to move the shape of an arpeggio up and down the fretboard in order to initiate, or because of, a change in key lends itself to being the primary choice of guitar players, helping ease the use of what is considered an intermediate technique of guitar performance.

In his presidential address he urged colleagues to adopt what he called " a middle line ": to combine the best of tradition with a fresh modern approach, to eschew dogma, and recognise " the influence of surroundings on the choice of materials and the technique of their use.

I Love Lucys pioneering use of three cameras led to it becoming the standard technique for the production of most sitcoms filmed in front of an audience, though single-camera remained the technique of choice for sitcoms that did not use audiences.

Delacroix's most influential work came in 1830 with the painting Liberty Leading the People, which for choice of subject and technique highlights the differences between the romantic approach and the neoclassical style.

" One must learn this art by practicing without technique — watching oneself: in daily activities ( walking, eating ), practices ( speech, gossip ), reactive emotions ( hate, jealousy )— becoming aware of these things " without any choice.

She also argues that given a choice, a human group will always choose the technique which has the lowest absolute labor cost rather than the highest yield.

The overall research showed that adolescents in music therapy “ change more when discipline-specific music therapy techniques, such as improvisation and verbal reflection of the music, are used .” The results of this study showed that music therapists should put careful thought and deliberation into their choice of technique with each individual client.

This technique is particularly useful on multiple choice problems.

Catheter-related infections may be minimised by appropriate choice of catheter and insertion technique.

In it they defended their freedom of choice in subject and technique, concluding:

The plant life available in a region affects the choice of material, which in turn influences the weaving technique.

Because of the discontinuous operating mode of those converters, a discontinuous sliding mode controller is a natural implementation choice over continuous controllers that may need to be applied by means of pulse-width modulation or a similar technique of applying a continuous signal to an output that can only take discrete states.