Page "Axiom of choice" ¶ 44
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## Some Related Sentences

However and ZF
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 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 ZF cannot prove that they are independent of ZF, even with the added hypothesis that ZF is consistent.
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 and plus
However, recent work has shown that for languages which make this distinction, all clicks have a uvular, or even pharyngeal, rear closure, and that the clicks explicitly described as uvular are in fact clusters / contours of a click plus a pulmonic or ejective component, in which the cluster / contour has two release bursts, the forward ( click ) and then the rearward ( uvular ) component.
However, the family Monachinae ( the lobodonts plus Monachus and Mirounga ) is probably paraphyletic.
However, a successful bid is worth the 10-point threshold plus the square of the bid, thereby rewarding a person bidding and making four tricks with 26 ( 10 plus 16 ) points.
However, no presidential candidate won the 50 % plus one vote majority stipulated in the constitution on the first round of voting.
However, the main attractions of the tour are three homecoming shows at Heaton Park, Manchester, on 29-30 June and 1 July 2012 plus one show in Dublin's Phoenix Park on 5 July 2012.
However, Arabic often avoids adverbs by using a cognate accusative plus an adjective.
However, a large, randomized, controlled trial ( the Women's Health Initiative ) found that women undergoing HT or HRT with conjugated equine estrogens ( Premarin ), in combination with a synthetic progestin ( medroxy pogesterone acetate ( Premarin plus Provera, known as Prempro )), had an increased risk of breast cancer and heart disease.
However, if a player chooses to draw so that 11 cards fit into melds, he or she can declare Big Gin in which case the player receives a Big Gin bonus of 31 points ( or another established amount ) plus any deadwood in the opponent's hand.
The 19th century author Aonghas MacCoinnich of Glasgow proposed that Scoti was derived from a Gaelic ethnonym ( proposed by MacCoinnich ) Sgaothaich from sgaoth " swarm ", plus the derivational suffix-ach ( plural-aich ) However, this proposal to date has not been met with any response in mainstream place-name studies.
However, Autolycus was a source of some controversy in Herakles ' life, because Autolycus stole some cattle from Euboea and Eurytus, who accused Herakles of the deed and, upon his going mad about these accusations, Herakles killed them plus another one of Autolycus ' sons, Iphitus.
However, in November 1981 ( after high-profile public demonstrations ) 40 frequencies unique to the UK, known as the 27 / 81 Bandplan using FM were allocated at 27 MHz plus 20 channels on 934 MHz ( 934. 0125 to 934. 9625 MHz with 50-kHz-spacing ).
However, the winner must win by 50 % plus one, called a simple majority.
However for cable television, it picks up Time Warner Cable of Albemarle, Stanly County which means that it only gets Charlotte television stations plus WFMY, the CBS affiliate in Greensboro.
However, after Shastri's death, by 1981, India had taken only 300, 000 Tamils as repatriates, while Sri Lanka had granted citizenship to only 185, 000 citizens ( plus another 62, 000 born after 1964 ).
However, with the advent of higher " Tiers of Service " of 10 Mbit / s downstream and higher, plus the reducing cost of NTL's cable modems ( supplied by Ambit Broadband ) NTL now supplies all subscribers with cable modems.
However, the plan was scaled back dramatically, particularly as only one night of primetime programming debuted as the network launched ; and by September 1995, The WB added only one additional night ( Sundays ), plus a three-hour Saturday morning and one-hour weekday morning children's block.
However, the Ultra loses its lead when anti-aliasing is enabled, because of the GeForce 3's new memory bandwidth / fillrate efficiency mechanisms ; plus the GeForce 3 has a superior next-generation feature set with programmable vertex and pixel shaders for DirectX 8. 0 games.
However, due to the high cost and inconvenience of using three separate graphics cards ( two Voodoo 2 SLI plus the general purpose 2D graphics adapter ), the Voodoo2 SLI scheme, though revolutionary at the time, had minimal effect on the total market share that the Voodoo2 held and was not a financial success.
However, Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.
However, the Sportshatch shared its bodyshell with the Viva Estate ( which was only ever produced in a two door plus tailgate configuration ).
However, additional construction projects, plus the expectation that the Parkway will pay for its own maintenance and policing ( and the massive E-ZPass project ) make it unlikely it will become toll-free in the foreseeable future.
However, concerns by the German Foreign Minister Baron Konstantin von Neurath and War Minister Field Marshal Werner von Blomberg that the pact might damage Chinese-German relations plus political disarray in Tokyo following the failed military coup of February 26, 1936 led to the Pact's being shelved for a year.
However, with trains frequently cancelled, and some timed to arrive at station after the start of the working day ( plus the evening workings also being timed to not be suitable for office workers ), patronage was very low and after a few years they were withdrawn.

However and GCH
However, in 2008, GCH Retail ( M ) Sdn Bhd closed the Giant Hypermarket and opened a Cold Storage supermarket outlet instead.

However and implies
However, the term dialect always implies a relation between languages: if language X is called a dialect, this implies that the speaker considers X a dialect of some other language Y, which then usually is some standard language.
However, while bitheism implies harmony, ditheism implies rivalry and opposition, such as between good and evil, or bright and dark, or summer and winter.
However, the term " farrier " implies a professional horseshoer with skill, education, and training.
However, this expression implies the ability to identify the particle with n < sub > 1 </ sub > as " particle 1 " and the particle with n < sub > 2 </ sub > as " particle 2 ".
However, James ' successor as leader of the Jerusalem church, Simeon, is not, in tradition, appointed till after the siege of Jerusalem in AD 70, and Eusebius ' notice of Simeon implies a date for the death of James immediately before the siege, i. e. about AD 69.
However, the law of the excluded middle is retained, because P and not-P implies P or not-P, since " or " is inclusive.
However, as Plutarch implies, since naval power relied on the mass mobilisation of the common citizens ( thetes ) as rowers, such a policy put more power into the hands of average Athenians — and thus into Themistocles's own hands.
However, the disadvantage this implies in relation to their contestant parties ( which denote permanent labor and a disciplined and organic structure ), impels them to modify their organization to become mass-based parties.
However, in order to reproduce the next stage of the argument, which eliminates the penultimate day of the week, the prisoner must argue that his ability to deduce, from statement ( A ), that the hanging will not occur on the last day, implies that a last-day hanging would not be surprising.
) However, since processor support for any SSE revision also implies support for MMX, the removal does not limit the types of data types usable by x86 SIMD.
However, this visible difference is merely a consequence of a deeper distinction: a DCT implies different boundary conditions than the DFT or other related transforms.
However, this term is also objectionable to some because it still implies that there is an issue, which they feel is created only by the PRC.
However, he implies later that he was lying about at least some of this information to mislead the Sontarans.
However, if the errors are not normally distributed, a central limit theorem often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large.
However, Willard's actions in checking the wraps of his opponent, not a normal procedure, implies he was concerned about foul play, Willard later claimed to have been defeated by gangsterism.
However, in a technical sense, the term " city manager ," as opposed to CAO, implies more discretion and independent authority that is set forth in a charter or some other body of codified law, as opposed to duties being assigned on a varying basis by a single superior such as a mayor.
However, Aristotle implies that a body is no longer the same body after death.
However, the information gleaned from recent scans of the fragments indicate that it may have come from the colonies of Corinth in Sicily and implies a connection with Archimedes.
However, this implies that the size of the observable universe is continually shrinking ; the distance to the edge of the observable universe which is moving away at the speed of light from any point moves ever closer.
However, this visible difference is merely a consequence of a deeper distinction: a DST implies different boundary conditions than the DFT or other related transforms.
However, if the value assigned to E < sub > ea </ sub > is negative, the negative sign implies a reversal of direction, and energy is required to attach an electron.
However, the majority of classical rabbis believed that the biblical phrase hallow the fiftieth year, together with the biblical promise that there would be three years worth of fruit in the sixth year, implies that the jubilee year was the 50th year.

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