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.

The choice of amine concentration in the circulating aqueous solution depends upon a number of factors and may be quite arbitrary.

The very nature of a choice so grounded in distinction and fact leads to the valid convictions which become force of will in the manifest leader.

The essence of contract is that one is free to make a choice of what one will or will not do.

If we fail to develop the means to hunt down and destroy the enemy's military force with extreme care and precision, and if war comes in spite of our most ardent desires for peace, our choice of alternatives will be truly frightening.

The inner panels do not have to be weatherproof, and the choice will depend on the quality of finish desired.

Sometimes, you have a choice of exposure for sites where the topography or trees of the area will provide afternoon shade, morning sun, or whatever may be most desirable for the use intended.

And with the shutdown method there will be no argument as to who gets the choice vacation dates.

Your choice of model will undoubtedly be governed by the number of people travelling in your group.

The fact seems to be that very many large branch stores are uneconomical, that the choice of location in the suburbs is as important as it was downtown, and that even highly suburbanized cities will support only so many big branches.

There may be other 1961 state committee retirements come April 18, but they will be leaving by choice of the Republican voters.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

We will abbreviate " Zermelo-Fraenkel set theory plus the negation of the axiom of choice " by ZF ¬ C.

His report of life there covers a wide range of topics, such as marriage in heaven ( where all angels are married ), children in heaven ( where they are raised by angel parents ), time and space in heaven ( there are none ), the after-death awakening process in the World of Spirits ( a place halfway between Heaven and Hell and where people first wake up after death ), the allowance of a free will choice between Heaven or Hell ( as opposed to being sent to either one by God ), the eternity of Hell ( one could leave but would never want to ), and that all angels or devils were once people on earth.

* 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.

As such, open theists resolve the issue of human free will and God's sovereignty by claiming that God is sovereign because he does not ordain each human choice, but rather works in cooperation with his creation to bring about his will.

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.

Where a driver will take you is partly up to his choice.

By an auspicious choice of the parameter n < sub > 0 </ sub >, we will arrive at a contradiction.

The Intermediates in the Class with the Judge were asked to pick 4 winners and give their reasons but their decisions did not affect the choice of the Judge.

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.

The choice does affect their starting position on the " Play on Earth " map, and thus different resources in one's initial cities, but has no effect on starting position when starting a random world game or a customized world game.

Philip Zimbardo discusses mind control as " the process by which individual or collective freedom of choice and action is compromised by agents or agencies that modify or distort perception, motivation, affect, cognition and / or behavioral outcomes ", and he suggests that any human being is susceptible to such manipulation.

While working there she gained her interest in Asia that would later affect her choice of fieldwork as a working anthropologist.

The choice of a coordinate system doesn't affect the properties of a vector or its behaviour under transformations.

A rapidly-growing mainstream literature from the 1970s has expanded beyond the model of economic policy in which planners maximize utility of a representative individual toward examining how political forces affect the choice of policies, especially as to distributional conflicts and political institutions.

It appears that the choice between a Long-Leaper and a Single-Leaper tends to affect game play by encouraging " hunkering down " and overdefending pieces, and allowing pieces to spread across the board more, with less attention to bulky blockades.

The choice of instruments, registers, and dynamics affect the overall tone color.

In addition to the choice of herbs and seasoning, the timing of when flavours are added will affect the food that is being cooked.

As this is an effective description of the system, it is a matter of choice as to what to define the experimental system to include and this will affect when " collapse " occurs.

Using their reasoning, he said, a choice of measurement setting here should not affect the outcome of a measurement there ( and vice versa ).

The choice and conditions of use of cryptographic mechanisms may affect card exportability.

The choice of EABI can affect performance.

Rotifers affect the species composition of algae in ecosystems through their choice in grazing.

A network administrator can reduce the cost of a spanning tree, if necessary, by altering some of the configuration parameters in such a way as to affect the choice of the root of the spanning tree.

Sunk costs should not affect the rational decision-maker's best choice.

As mentioned above, choice of individual assembler instructions ( or even sometimes just their order of execution ) on particular machines can affect the efficiency of an algorithm.

Over a two-week treatment of healthy volunteers, sertraline slightly improved verbal fluency but did not affect word learning, short-term memory, vigilance, flicker fusion time, choice reaction time, memory span, or psychomotor coordination.

The choice of deck does not affect the rules.

The choice of transition may affect the amount of inter-cache traffic, which in turn may affect the amount of cache bandwidth available for actual work.

While this may affect the choice of an antenna, an antenna's impedance can also be adapted to the desired impedance level of a system using an matching network while maintaining the other characteristics ( except for a possible loss of efficiency ).

Over concern about side-effects, for example weight gain and acne, often affect choice.