Without the axiom of choice, there might exist cardinalities incomparable to ( namely, the cardinalities of Dedekind-finite infinite sets ).

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 these massive trucks highways would still be just an idea of the future.

Without these APIs, the programmer needs to write these APIs him / herself.

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

" 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 any such logical operators of valence 0, these two constants can only be expressed using quantifiers.

Without these precautions, cracks in overgrown and overly brittle hoof walls are a danger, as is bruising of the soft tissues within the foot because of inadequately thick and hard sole material.

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 these crutches, reading a Kvikkalkul program requires one to work out what the program actually does.

Without glacial sources, annual flow regimes in these rivers are more variable although limited flow persists through the dry season.

Without these two events, Italian unification would have been a more prolonged process.

Without these features, they are usually noncompliant.

Without at least one of these two pieces of information, we are not obviously justified in seeing design in such cases … he problem for the fine-tuning argument is that we lack both of the pieces that are needed to justify an inference of design.

Without water, these particular metabolic processes could not exist.

Without appropriate intervention, these children lag behind their peers.

Without the transfer of these rights, Tufts could come in at any time and demand the removal of buildings to obtain minerals.

Without the bright flame and dense smoke trail produced by the burning of aluminized propellants, these smokeless propellants all but eliminate the risk of giving away the positions from which the missiles are fired.

Without good flexing properties these movements rapidly form cracks and, ultimately, make the rubber article fail.

Without these opaque layers, the hot, luminous core emits ultraviolet radiation that ionizes the ejected outer layers of the star.

Without halftoning, the three primary process colors could be printed only as solid blocks of color, and therefore could produce only seven colors: the three primaries themselves, plus three secondary colors produced by layering two of the primaries: cyan and yellow produce green, cyan and magenta produce a purplish blue, yellow and magenta produce red ( these subtractive secondary colors correspond roughly to the additive primary colors ) plus layering all three of them resulting in black.

Without social capital in the area of education, teachers and parents that play a responsibility in a students learning, the significant impacts on their child ’ s academic learning can rely on these factors.

Without sufficient levels of these key electrolytes, muscle weakness or severe muscle contractions may occur.

Without trying to make a direct analogy one can be convinced of the importance of considering these extra properties following the rationale implied by this game: an object is defined here consisting of two rods and strings connecting them.

Without timely treatment, these new blood vessels can bleed, cloud vision, and destroy the retina.

Without this power we lay open any American industry which the Europeans may find it economically profitable to destroy to the will of others.

Without such precautions, most programs may compile only on a certain platform or with a particular compiler, due, for example, to the use of non-standard libraries, such as GUI libraries, or to the reliance on compiler-or platform-specific attributes such as the exact size of certain data types and byte endianness.

Without any effort, Munch had a steady stream of female models, some of whom he may have had sexual relations with, and who were the subjects of numerous nude paintings.

A curriculum-free philosophy of homeschooling may be called unschooling, a term coined in 1977 by American educator and author John Holt in his magazine Growing Without Schooling.

Without specific recourse to any specific representation of this deity Murray speculates that the head coverings common in inquisition-derived descriptions of the devil ' may throw light on one of the possible origins of the cult '.

Without careful moisture and temperature control during manufacture and packaging, the tea may become unfit for consumption, due to the growth of undesired molds and bacteria.

Without the resonator ( which may be as simple as a table top to which the handle is pressed ), the sound is very faint.

Without a robust whistleblower system, directors / trustees of the organization may fail in their oversight responsibilities.

: Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong.

Without this charge and delay, the motor may ignite a second stage ( black powder only ).

Without this information, prospective buyers and insurance appraisers may not take the collector's word.

Without this device, workers may overtighten the micrometer on the work, causing the mechanical advantage of the screw to squeeze the material or tighten the screw threads, giving an inaccurate measurement.

* Without a water tower, parts of gravity flow water supply systems in hilly areas may be subject to negative pressures ( see siphon ).

Without collegiate training directing her research, Goodall observed things that strict scientific doctrines may have overlooked.

Without sufficient dissolved oxygen in the water, animals and plants may die off in large numbers.

Without the ability to swap genes with its pair, it is racking up with mutations and may stop functioning in 10 million years.

Without the use of a pressure cooker, boiled foods may be undercooked, as described in Charles Darwin's The Voyage of the Beagle ( chapter XV, March 20, 1835 ):

Without treatment, episodes of acute gout may develop into chronic gout with destruction of joint surfaces, joint deformity, and painless tophi.

* For integrity: Without the hash function, the text " to be signed " may have to be split ( separated ) in blocks small enough for the signature scheme to act on them directly.

Without the use of objective recording techniques, it may be very difficult to distinguish between these conditions.

Without the first layer of " capture " antibody, any proteins in the sample ( including serum proteins ) may competitively adsorb to the plate surface, lowering the quantity of antigen immobilized.

* Reporters Without Borders considers the number of journalists murdered, expelled or harassed, and the existence of a state monopoly on TV and radio, as well as the existence of censorship and self-censorship in the media, and the overall independence of media as well as the difficulties that foreign reporters may face.

Without either Toxin A or Toxin B, C. difficile may colonize the gut but is unlikely to cause pseudomembranous colitis.