Because of independence, the decision whether to use of the axiom of choice ( or its negation ) in a proof cannot be made by appeal to other axioms of set theory.

Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, e. g., in the proof of book IX, proposition 20.

Because statements of a formal theory are written in symbolic form, it is possible to mechanically verify that a formal proof from a finite set of axioms is valid.

Because of their use of natural language, the standards of rigor for informal proofs will depend on the audience of the proof.

Because no guild records were kept prior to 1494 in Leuven there is no concrete proof that Matsys attained his master's status there ; however, historians generally accept this to be the location of his early training because there he had not been previously registered in Antwerp as an apprentice.

Because of the political position of George Clinton as Vice President at the time of Burrville's name change, compared to DeWitt Clinton's position as the mayor of New York City, most likely the residents of the town of Burrville would have been more readily identifiable and more honorable toward George Clinton than DeWitt ; therefore, it is most likely Clinton was named after George Clinton, barring historical proof.

Because if not, then an elementary proof of Euclid's result is also impossible.

Because the proof uses only well-accepted methods of deduction, and because none of these methods appears to be incorrect, this situation is paradoxical.

Because an affirmative defense requires an assertion of facts beyond those claimed by the plaintiff, generally the party who offers an affirmative defense bears the burden of proof.

Because it is difficult to determine exactly when a quorum was lost, points of order relating to the absence of a quorum are " generally not permitted to affect prior action ; but upon clear and convincing proof, such a point of order can be given effect retrospectively by a ruling of the presiding officer, subject to appeal.

Because of this, derivability is stable under additions to the proof system, whereas admissibility is not.

Because " proof " refers to the process by which the coins are made and not the condition of the coin certification agencies can grade and assign numerical ratings for proof coins.

Because of the extremely high burden of proof on the plaintiff, and the difficulty in proving essentially what is inside a person's head, such cases — when they involve public figures — rarely prevail.

Because of soundness, we know that the user must really act honestly in order to be able to provide a valid proof.

Because of zero knowledge, we know that the user does not compromise the privacy of its secrets in the process of providing the proof.

Because the standards of proof were so high, it was well-nigh impossible to inflict the death penalty.

Because ALS laws are immedediate and require no proof of guilt, proponents such as Mothers Against Drunk Driving argue that they are more effective in reducing drunk driving than are traditional post-conviction sanctions, and that, in any event, driving is only a privilege.

Because of this, no proof could be found to show that the odd zeta constants were irrational, even though they were — and still are — all believed to be transcendental.

: Because the State has greater access to information concerning customary flight patterns and because the coercive power of the State ought not be brought to bear in cases in which it is unclear whether the prosecution is a product of an unconstitutional, warrantless search, the burden of proof properly rests with the State and not with the individual defendant.

Because title to property that arises by operation of law is usually contingent upon proof of certain contingencies, and title records may not contain evidence of those contingencies, legal proceedings are sometimes required to turn title that arises by operation of law into marketable title.

Because the Court ordinarily defers to the legislature's stated intent, only the clearest proof will suffice to override that intent and transform what has been denominated a civil remedy into a criminal penalty.

Because the WD1770 is capable of single density mode and uses the same IBM360 derived floppy disc format as the Intel 8271 found in the BBC Micro, it was also possible to run a DFS filing system with an alternate ROM, such as the P. R. E. S AP4 interface.

Because field artillery mostly uses indirect fire the guns have to be part of a system that enables them to attack targets invisible to them in accordance with the combined arms plan.

Because the Doom engine uses the binary space partitioning system for rendering, it does not enable moving walls.

Because it relies on volunteers and uses " rough consensus and running code " as its touchstone, results can be slow whenever the number of volunteers is either too small to make progress, or so large as to make consensus difficult, or when volunteers lack the necessary expertise.

Because LEDs emit less heat than incandescent bulbs, they are an energy-efficient technology for uses such as in freezers and refrigerators.

Because of the scarcity, high radioactivity and high toxicity, there are currently no uses for protactinium outside of scientific research, and for this purpose, protactinium is mostly extracted from spent nuclear fuel.

Because the body uses gastric acid to release B < sub > 12 </ sub > from food particles, decreased vitamin B < sub > 12 </ sub > absorption may occur with long-term use of proton-pump inhibitors and may lead to Vitamin B12 deficiency.

Because of its depth and relative ease of access, the US Navy uses Seneca Lake to perform test and evaluation of equipment ranging from single element transducers to complex sonar arrays and systems.

Because it uses frequency modulation, SECAM is not linear with respect to the input image ( this is also what protects it against signal distortion ), so electrically mixing two ( synchronized ) SECAM signals does not yield a valid SECAM signal, unlike with analog PAL or NTSC.

Because Adorno believed that sociology needs to be self-reflective and self-critical, he believed that the language the sociologist uses, like the language of the ordinary person, is a political construct in large measure that uses, often unreflectingly, concepts installed by dominant classes and social structures ( such as our notion of " deviance " which includes both genuinely deviant individual and " hustlers " operating below social norms because they lack the capital to operate above: for an analysis of this phenomenon, cf.

Because nouns and adjectives share these three categories, Dionysius Thrax does not clearly distinguish between the two, and uses the term ónoma for both, although some of the words that he describes as paragōgón ( pl.

Because multipath propagation behaves differently in each kind of wire, G. hn uses different OFDM parameters ( OFDM symbol duration, Guard Interval duration ) for each media.

Because of its unusual properties, molasses has several uses beyond that of a straightforward food additive.

Because D uses garbage collection, in both versions, there is no need to track usage of variables as they are passed.

Because ASCII in general does not support overstrikes in a device-independent way, and does not include a division symbol per se, J uses % to represent division, as a visual approximation or reminder.

Because most high energy gamma photons will pass through the low density fill gas without interacting, the tube uses the interaction of photons on the molecules of the wall material to produce high energy secondary electrons within the wall.

Because counting sort uses key values as indexes into an array, it is not a comparison sort, and the Ω ( n log n ) lower bound for comparison sorting is inapplicable to it.

Because it uses advanced propulsion technology and standard track gauge ( unlike the rest of Toronto's system ), it can not interchange with Toronto's subway or streetcar lines.

Because color TVs were so common between the 1960s and 2000s, economies of scale have driven down the cost of colorburst crystals, which often find uses in various other applications, such as Ham radio ( 3. 579 Mhz is common QRP calling frequency at 80m band ).

Because of this basis in human visual perception, Munsell ’ s system has outlasted its contemporary color models, and though it has been superseded for some uses by models such as CIELAB ( L * a * b *) and CIECAM02, it is still in wide use today.

Because Louisiana has a system that, uniquely among the states, uses civil law, the Paul M. Hebert Law Center at Louisiana State University in the now offers a joint J. D ./ Diploma of Civil Law ( D. C. L.

Because the IOS needs to know the cleartext password for certain uses, ( e. g., CHAP authentication ) passwords entered into the CLI by default are weakly encrypted as ' Type 7 ' ciphertext, such as "< tt > Router ( config )# username jdoe password 7 0832585B1910010713181F </ tt >".

Because of this, the internal workings of an object may be changed without affecting any code that uses the object.