However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

However, ZF plus GCH implies AC, making GCH a strictly stronger claim than AC, even though they are both independent of ZF.

However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.

However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF ( a Grothendieck universe ), and its subsets can be thought of as " classes ".

" However, proponents of more limited forms of constructive mathematics would not assert that ZF itself is a constructive system.

However, κ does not need to be inaccessible, or even a cardinal number, in order for V < sub > κ </ sub > to be a standard model of ZF ( see below ).

However, seven-speed autos are becoming available in some cars ( found in Mercedes 7G gearbox, Infiniti ), as are eight-speed autos in models from 2006 introduced by Aisin Seiki Co. in Lexus, ZF and Hyundai Motor Company.

However, ZF ( 5HP590 ) and Voith transmissions were also available on some models.

However, it is not uncommon to talk about inner models of subtheories of ZFC ( like ZF or KP ) as well.

However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF.

However, some of these sets contain subclasses that are not sets, which makes them different from Cantor ( ZF ) finite sets and they are called infinite in AST.

However, if there are infinitely many Woodin cardinals with a measurable above them all, then L ( R ) is a model of ZF that satisfies AD.

However, it cannot be presumed, informal though the structure of the population seems, that there are not well-defined roles within the system.

However, the building of freeways and garages cannot continue forever.

However, since no equipment existed to show the images in motion, such a series of images cannot be called animation in a true sense of the word .< ref >

" However, their doctrines cannot be considered representative of traditional Arian doctrines or vice-versa.

However, as the preservation of behaviour in the fossil record is exceedingly rare, these ideas cannot readily be tested.

However, many later disapproved of his actions, including to an extent Machiavelli, who claimed " It cannot be called prowess to kill fellow-citizens, to betray friends, to be treacherous, pitiless, irreligious.

However, beauty cannot be reduced to any more basic set of features.

However, there is limited evidence that mistletoe's effects on the immune system help the body fight cancer .... At present, the use of mistletoe cannot be recommended outside the context of well-designed clinical trials.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

However, invalid formulas ( those that are not entailed by a given theory ), cannot always be recognized.

However, in a supersonic flow, the pressure disturbance cannot propagate upstream.

However, God tells Judah ( through Isaiah ) that the covenant cannot protect them when they have broken it by the worship of other gods and by acts of injustice and cruelty, which oppose God's law.

However, if the document does not specify the encoding either, this may result in the equally bad situation where the user agent displays mojibake because it cannot find out which character encoding was used.

However, current research cannot ascertain whether the extinctions occurred prior to or during the boundary interval itself.

However in Daoism it refers more often to a meta-physical term that describes a force that encompasses the entire universe but which cannot be described nor felt.

However, another observer cannot benefit until the results of that measurement have been relayed to him, at less than or equal to the speed of light.

However, UCI regulations stipulates a legal race bike cannot weigh less than 6. 8 kg ( 14. 99 lbs ).

However, this assembly cannot collapse to a size dictated by the volumes of these individual particles.

However, the odds bet cannot be made independently, so the house retains an edge on the pass line bet itself.

However, an understanding of the way words in the two languages evolve from Proto-Indo-European ( PIE ) roots shows that they cannot be cognate ( see for example Grimm's law ).

From such a surrender, the dissolution of the body corporate ensues .” Nor does there seem to have been much question that by “ a judgment of forfeiture against a corporation itself, it may be dissolved .” However, Supreme Court Justice Wilson, lecturing in his unofficial capacity, at least, suggests his displeasure with the doctrine that corporate dissolution cannot be predicated “ by a judgment of ouster against individuals.

However, as per rules of Islamic Fiqas, Halal ( permitted ) from Muhammad cannot be altered by any authority or individual.

However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable.

However, these high doses cannot be given because they would be fatal to the patient.

However, he cannot permanently ' reprogram ' human minds to believe what he might want them to believe even if he wanted to do so, explaining that the mind is an organism that would always recall the steps necessary for it to reach the present and thus ' rewrite ' itself to its original setting if he tried to change it.

However, in the late sixties Roger Penrose and Stephen Hawking used global techniques to prove that singularities appear generically.

However fame based on one program may often prove short-lived after a program is discontinued.

However, it is a severely limited conclusion that does nothing to prove that one's own body exists, let alone anything else that is perceived in the physical universe.

However, Biafra claims the above mentioned licensing deals prove otherwise.

However, if all of the different toothpastes are good and equal, there is no need to prove their claim.

However the move was to prove disastrous and was followed by a series of natural disasters including the flooding of half the city.

However, there would be some bright spots that would prove to be good investments in the near future.

However, as the place did not prove to be defensible, they settled in the remains of an older town upon a hill not far away and founded a new town, which they named Tábor ( after the traditional name of the mountain on which Jesus was expected to return ; see Mark 13 ); hence they were called Taborites.

However, the manner in which the injured person could seek to prove those things is quite variable.

However, his interest in theosophy did not prove to be enduring and he left the society shortly afterwards Brooks departed as his tutor.

However, it is useful in generating corrections of astronomical models, to prove of long-termed stability and of avoiding of orbital resonance in solar system.

However, the realization that eye movements performed in dreams may affect the dreamer's physical eyes provided a way to prove that actions agreed upon during waking life could be recalled and performed once lucid in a dream.

However, he also acknowledged that, while greatly improving his trumpet playing technique, Juilliard helped give him a grounding in music theory that would prove valuable in later years.

However these negotiations may prove difficult in that both parties in a conflict believe that they are morally superior to the other, and are unwilling to negotiate on basis of moral equivalence.

However, for many NLP problems, the definition of a gold standard is a complex task, and can prove impossible when inter-annotator agreement is insufficient.

However, microfossils such as stromatolites and oncolites prove that in shallow marine environments at least life did not suffer any perturbation.

However, so long as the owner of the trade secret can prove that reasonable efforts have been made to keep the information confidential, the information remains a trade secret and generally remains legally protected.

However, all people are influenced and act as influences, and ironically, the book itself may influence its reader, though the preface paradoxically states that no artist, in their work, " desires to prove anything " or has " ethical sympathies ".

However, Arnold's books have been criticized for supporting the theory with statements meant to teach an intuitive understanding, without providing the tools necessary to prove these statements.

However, Dagmar served a six-month prison sentence for consorting with an apparently German officer: thinking that Chapman was dead, she was unable to prove that he was a British agent.

However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles ( otherwise, for example, all rhombi would be similar ).

However, he did make it known that he would not decline the GOP nomination if it were offered to him, and some conservative Republicans hoped that by winning a primary contest he could prove his popularity with voters.

However, if in three to six months of efforts at remedying risk factors prove insufficient, then drug treatment is frequently required.

However, the law would also prove to be a major political liability for Arthur.

However they dropped this charge in October 2008, but maintain they could prove the charge and were only dropping the charge to expedite proceedings.