Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.

Without this axiom, we simply have the axioms which define a totally ordered field, and there are many non-isomorphic models which satisfy these axioms.

Without the axiom of choice, there would be cardinal numbers which were not well-orderable.

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 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 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 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 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 ; ;

Without a system of values there can be no society.

Without observational constraints, there are a number of candidates, such as a stable supersymmetric particle, a weakly interacting massive particle, an axion, and a massive compact halo object.

Without it, household behavior would be unaffected by uncertain employment and income prospects, financial and capital markets would reduce to exchange of a single instrument in each market period, and there would be no communications industry.

" 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 Coleman, Essendon's fortunes plummeted, and there were to be no further premierships in the 1950s.

Without understanding, there cannot be true faith and that understanding is built on the foundation of the community of believers, the scriptures and traditions and on the personal experiences of the believer.

Without it, there is no flamenco.

Without quantum mechanics, there would be no diamagnetism, paramagnetism or ferromagnetism.

* Without Colors — Before there was an atmosphere, everything was the same shade of gray.

Without an understanding of Puritanism … there is no understanding of America.

Without knowledge of the Bible there can be no peace in the life of the Church or of society, and outside of it there is no real and abiding good ; it is the one authority for the faith.

In Van Buren's own words, " Without strong national political organizations, there would be nothing to moderate the prejudices between free and slaveholding states.

Without the field, there are electrons moving equally in all directions.

Near the end of her time as governor general, Adrienne Clarkson stated: " My constitutional role has lain in what are called ' reserve powers ': making sure that there is a prime minister and a government in place, and exercising the right ' to encourage, to advise, and to warn '[...] Without really revealing any secrets, I can tell you that I have done all three.

Without phenotypic variation, there would be no evolution by natural selection.

Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved.

Without hesitation, Truman picked Marshall, adding " I don't think in this age in which I have lived, that there has been a man who has been a greater administrator ; a man with a knowledge of military affairs equal to General Marshall.

Without this there was a risk of arcing which would burn the insulation between the contacts.

Without a meaning in life, there is no scale of values.

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 the door stop damage might be done to the wall.

Without his support, the senate might have remained too frightened to even meet.

Without some earlier ones, such as the spinning jenny and flying shuttle in the textile industry and the smelting of pig iron with coke, these achievements might have been impossible.

Without Virginia, a new convention might have been held and a new constitution written in a much more polarized atmosphere, since the constitution did not specify what would happen if it was only partially ratified.

Without Stevenson, Sterne might have been a more decorous parish priest, but then might never have written Tristram Shandy.

Without Thuggee's sacred service, Kali might destroy all the human kind:

Without such instrumentation one might have to have someone crouched over Windows Task Manager at the server to see how much CPU load the performance tests are generating ( assuming a Windows system is under test ).

Without great care, the cable might part and would be difficult to repair.

Without such advertisement, a car-thief might, for example, approach a car, break the window, and then flee in response to an alarm being triggered.

Without the Seven Years ' War to " unite " the two countries briefly, the marriage of Maria Antonia and the Dauphin Louis-Auguste might not have occurred.

Without it, there might not have been either That Was the Week That Was or Private Eye magazine, which originated at the same time, and that partially survived due to financial support from Peter Cook, and served as the model for the later American Spy Magazine.

Without careful analysis of the definition, one might erroneously conclude that an instance of http :// www. w3. org / TR / 2004 / REC-owl-guide-20040210 / wine # Merlot was something physical, instead of a type of wine.

In The Musical Quarterly, Martial Teneo and Theodore Baker wrote, " Without the example set by Hervé, Offenbach might perhaps never have become the musician who penned Orphée aux Enfers, La belle Hélène, and so many other triumphant works.

Without Bonnie, the media outside Texas might have dismissed Clyde as a gun-toting punk, if it ever considered him at all.

Without his prestige the Angevin dynasty might not have survived the disastrous reign of John ; where the French and the rebels would not trust the English king's word, they would trust William.

Without any additional assumptions, this task cannot be solved exactly since unseen situations might have an arbitrary output value.

Without the 8-9 swap, Conrad might have commanded Apollo 11, which became the first mission to land on the Moon.

Without such information, market researchers and advertisers, with their expertise, could generalize, on the basis of a concept test, as to how consumers might react to the actual products or to variations of the tested concepts.

Three other works that might have been Rowley solo plays have not survived: Hymen's Holidays or Cupid's Vagaries ( 1612 ), A Knave in Print ( 1613 ), and The Fool Without Book ( also 1613 ).

" Without resolution, Thoreau used " his retreat to the woods as a way of framing a reflection on both what ails men and women in their contemporary condition and what might provide relief.

Without a constant influx of precious metals from an outside source, and with the expense of continual wars, it would seem reasonable that coins might be debased to increase the amount that the government could spend.