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 >).

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.

" Without the help of Doña Marina ," he writes, " we would not have understood the language of New Spain and Mexico.

Without Golding's translation, common knowledge of the ancient text would not have been understood in the same clear light.

Without doubt its implementation is better understood thanks to the critiques that both Kuhn and Feyerabend have made in response to certain theses proposed by followers of the received view of theories.

Without an understanding of quantum gravity, a theory unifying quantum mechanics and relativistic gravity, the physics of the Planck epoch are unclear, and the exact manner in which the fundamental forces were unified, and how they came to be separate entities, is still poorly understood.

“ Act Without Words II shows that life must be endured, if not understood.

Without careful qualification this can be misleading.

Without the good magazines, without their book reviews, their hospitality to European writers, without above all their awareness of literary standards, we might very well have had a generation of Krim's heroes -- Wolfes, Farrells, Dreisers, and I might add, Sandburgs and Frosts and MacLeishes in verse -- and then where would we be??

Without public scrutiny the deliberations of public agencies would no doubt be conducted more speedily.

Without a great acceleration in the metropolitan area's economy, there will not be sufficient jobs for the growing numbers of youngsters, and St. Louis will slip into second-class status.

Without question, this time and temper saver should be immediately installed at the Woodside station.

Without these massive trucks highways would still be just an idea of the future.

Without a system of values there can be no society.

Without any definite plan in mind, she went to a judge to see what could be done.

Without RIMAs or the MAOI of MAO-A, DMT would be oxidised ( and thus rendered biologically inactive ) by monoamine oxidase enzymes in the digestive tract.

Without a register like an accumulator, it would be necessary to write the result of each calculation ( addition, multiplication, shift, etc.

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 documents, they could not be employed, educated, treated in hospitals, travel outside of the country, or vote, among other hardships.

Without this double feedback, the system would still be bistable, but would not be able to tolerate such a wide range of concentrations.

Without knowledge of the key, it should be difficult, if not nearly impossible, to decrypt the resulting ciphertext into readable plaintext.

Without a significant amount of memory, a computer would merely be able to perform fixed operations and immediately output the result.

Without the deactivation of the catalyst, the ethene produced will be further reduced to ethane.

Without much added cost, these devices could be designed to also harvest BSFL.

Without them, they would be unable to obtain any nutritional value from plants.