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Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers ( which are not constructively equivalent to the Cauchy real numbers without countable choice < ref > For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman ; 1998 ; < cite > A weak countable choice principle </ cite >; available from .</ ref >).
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Without and using
Without using the term, 1920 saw the first rule that would be referred to today as defensive indifference, as stolen bases would not be credited, unless an effort was made to stop the runner by the defense.
Without this restriction, bizarre maps ( using regions of finite area but infinite perimeter ) can require more than four colors.
Without any such logical operators of valence 0, these two constants can only be expressed using quantifiers.
Without first finding the radian frequency, the natural frequency can be found directly using:
Without using overscan, the Amiga's graphics display is 320 or 640 pixels wide by 200 ( NTSC ) or 256 ( PAL ) pixels tall.
Without using measures of income adjusted for inflation, they will tend to overstate the effects of economic growth.
Without copying any one historical model, and without compromising its authentic stone-on-stone construction by using modern steel girders, Saint John the Divine is an example of the 13th century High Gothic style of northern France.
The use of phonics in American education dates at least to the work of Favell Lee Mortimer, whose works using phonics includes the early flashcard set Reading Disentangled ( 1834 ) and text Reading Without Tears ( 1857 ).
* Without using RMTI, Hop count can not exceed 15, in the case that it exceeds this limitation, it will be considered invalid.
* Without using RMTI, RIP has slow convergence and count to infinity problems.
Without actually accusing the group of using the hashish drug, the Caliph used the term in a pejorative manner.
Without an extra Global navigation satellite system, receiver tracking is difficult, but not impossible, as the position of a mobile unit can be determined using a Doppler shift calculation from the satellite.
Without analyzing the statistical uncertainty of a given 100-year flood, scientists and engineers can decrease the uncertainty by using two practical rules.
Without warning, in 2000 French urban climber Alain " Spiderman " Robert, using only his bare hands, climbing shoes and no safety devices, scaled the obelisk all the way to the top.
Without using an IF, all the complicated filters and detectors in a radio or television would have to be tuned in unison each time the frequency was changed, as was necessary in the early tuned radio frequency receivers.
Without using their considerable hydropower output, silver mining in the Harz would never have been able to attain its major economic significance.
Without these local organizations pushing for their rights and, many times, using self-defense tactics, not much would have changed, according to scholars such as Hill.
Without the time or resources to build sufficient aircraft and train experienced pilots, the surviving Japanese carriers were mere shadows of their former selves, a fact the Japanese recognized by using them as sacrificial decoys at Leyte Gulf.
Kean has also been criticized for using his role as the chairman of the 9 / 11 Commission in order to make profit, such as his book, Without Precedent.
* Without taking more water, wipe your hair from the middle down to the forehead, or vice versa, using the index finger of your right hand ; it is mustahab to use three fingers.
Without the software installed, the F keys will do nothing, and toggling them back to normal by using the F Lock has been found by some to be unreliable as the keyboard will often randomly and inexplicably " untoggle " the F Lock key.
Without all-seeing devices like radar, a pilot could approach his foe stealthily, using clouds, haze or even using the enemy aircraft's own wings or tail to conceal his approach.
Without a case to provide these functions, caseless rounds using nitrocellulose will begin to cook off, firing from the residual chamber heat, much sooner than metal cased cartridges would.
Without and countable
Without this suffix, a countable noun is understood to be singular.
Without and choice
Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.
Without the axiom of choice, there might exist cardinalities incomparable to ( namely, the cardinalities of Dedekind-finite infinite sets ).
Without such amenities as Chien-Tan campus, students at Ocean Campus often had no choice but to use their greater opportunities and free time at Ocean campus to communicate and bond.
Without the support of the US, the anticommunist elements in the government had little choice other than to gradually allow the Pathet Lao to take power.
The statement read in part, " Without the specific changes we are seeking across the entire Chronicle organization, we will have no choice but to quickly seek a buyer for The Chronicle or, should a buyer not be found, to shut the newspaper down.
Without the axiom of choice, there would be cardinal numbers which were not well-orderable.
Without much choice, Shankar signed a newcomer Bala to take over the set design and art work for the film.
Without another choice, Kirk and the others plan strategies and begin to manufacture primitive spears.
Without too much math we can write probability of choice statements based on attractiveness, and these take a form similar to some varieties of disaggregate demand models.
Without realizing it, the land they had purchased was actually the church and monastery of Saint Matthew, the site which was the alleged choice of the Virgin to enshrine the icon.
Without fixed ropes to retreat down in case of emergency, or a lower camp to return to, the commitment of alpine style is greater than expedition style in terms of the choice to ascend or descend.
Without an emphasis on personal choice, one may make use of an external moral system as a tool to moralize otherwise immoral acts, leading to negation of the self.
Without effective input at a senior level to correct for male bias, she asserts that historical attitudes towards women's physiology limits them in their choice of career, their intellectual maturity, their credibility and their ability to be effective contributors to the advancement of human society.
Without Warren to prove that they weren't behind the kidnapping, William and Aren have no choice but to rescue Farril themselves.
* Without loss of generality-in a mathematical proof, proving for one of several cases where the choice of case does not matter
Without and is
Without saying or seeming to say that in portraying the Sartoris and the Compson families Faulkner's chief concern is social criticism, we can say nevertheless that through those families he dramatizes his comment on the planter dynasties as they have existed since the decades before the Civil War.
Without a precise knowledge of Germanic philology, however, it is debatable whether their use was not more often a source of confusion and error than anything else.
Without such forward planning, investment funds are wasted because manufacturing facilities are completed before there is power to operate them or before there is transport to service them ; ;
A true university, like most successful marriages, is a unity of diversities Without forcing all components into a single pattern, the preparation of a master plan is an opportunity to consider interrelation of knowledge at its highest level, which a university -- in contrast to a multiversity -- should stand for.
Without losing the distinctive undertow of Brahmsian rhythm, the pacing is firm and the over-all performance has a tightly knit quality that makes for maximum cumulative effect.
`` Without any officious and improper interference on the subject, the price of labor or the wages of mechanics will be regulated by the demand for the manufactured article and the value of that which is paid for it ; ;
Without such a declaration, Paine concluded, “ he custom of all courts is against us, and will be so, until, by an independence, we take rank with other nations .”
Without this money creation benefit, it is actually more expensive to outsource the IT operations as the outsourcing adds a layer of management and increases overhead.
Chapter 17 of William Bates ' 1920 book Perfect Sight Without Glasses, in which the author argues that observation of the sun is beneficial to those with poor vision, includes a figure of somebody " Focussing the Rays of the Sun Upon the Eye of a Patient by Means of a Burning Glass.
Without a freewheel, coasting is impossible, so when the rear wheel is moving, the cranks are moving.
Without diagnosis and treatment, the infection can be fatal in one to six days ; mortality in untreated cases is approximately 100 %.
Without clear documentation, software can be hard to use — especially if it is very specialized and relatively complex like Photoshop or AutoCAD.
Without protected memory, it is possible that a bug in one program will alter the memory used by another program.
Without salvation from sin ( see below ), a person's separation from God is permanent, causing such a person to enter Hell in the afterlife.
Guided by Arkyn, Corum, Rhalina and companion Jhary-a-Conel cross the planes and encounter the King Without A Country, the last of his people who in turn is seeking the City in the Pyramid.
According to Tibor R. Machan, " Without a market in which allocations can be made in obedience to the law of supply and demand, it is difficult or impossible to funnel resources with respect to actual human preferences and goals.
: Without doubt this is the most important work on general algebra that the Annalen has ever published.
Without experience of meditation, one's knowledge of the world is too limited to fully understand dukkha, as required by the first noble truth, and proceed to enlightenment.
" Without these questions there is no clear fulcrum on which to balance law, politics, and the practice of arbitration — in fact, no common assumptions of all participants — so the ability to formulate the questions are prior to rights balancing.
Without a price mechanism, socialism lacks the means to relate consumer satisfaction to economic activity, it is argued.
The scene in which Lee states that his style was the style of " Fighting Without Fighting " and then lures Parsons into boarding a dinghy is based upon a famous anecdote involving the 16th century samurai Tsukahara Bokuden.
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