It is also consistent with ZF + DC that every set of reals is Lebesgue measurable ; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption ( the existence of an inaccessible cardinal ).

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself.

Pan ( 1986 ) proved an lower bound assuming a bound on a measure of the FFT algorithm's " asynchronicity ", but the generality of this assumption is unclear.

When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivism.

In the letter, Germain claimed to have proved the theorem for n = p – 1, where p is a prime number of the form p = 8k + 7 ; however, her proof contained a weak assumption.

In the same way we have to reduce ( reductionism ) each assumption and belief to something that can be proved through the self-evident reason ( empirically verified ).

The fallacy of petitio principii, or " begging the question ", is committed " when a proposition which requires proof is assumed without proof ", or more generally denotes when an assumption is used, " in some form of the very proposition to be proved, as a premise from which to deduce it ".

Decades later John Bell proved Bell's theorem ( see p. 14 in Bell ), in which he showed that, if they are to agree with the empirical predictions of quantum mechanics, all such " hidden-variable " completions of quantum mechanics must either be nonlocal ( as the Bohm interpretation is ) or give up the assumption that experiments produce unique results ( see counterfactual definiteness and many-worlds interpretation ).

Frankel proved that no Jewish doctrine justified such an assumption, and owing to his work a new regulation ( 13 February 1840 ) put the Jews on the same basis as Christians as regards testimony in court.

While Cauchy felt that the assumption was minor, most historians believe that the proof was not complete until Abel proved this assumption.

Its goal was the closing of suspected insurgent supply routes of volunteers and material from Syria, and with the fight they received their assumption proved correct.

The new president of UEFA, Michel Platini, was reported to be in favour of expansion which proved an accurate assumption.

The scale and duration of the battle was minor compared to later fighting but the First Battle of Krithia was one of the most significant of the campaign as it proved that the original British assumption of a swift victory over an indifferent enemy was grossly mistaken.

It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results.

Much of his writing was the kind of " Boy's books " initiated by the famous Stratemeyer Syndicate, based on the assumption ( which proved hugely successful ) that " boys want the thrill of feeling ' grown-up '" and that they like books which give them that feeling to come in series where the same heroes appear again and again.

For example, the computer language Prolog uses a sort of default assumption when dealing with negation: if a negative atom cannot be proved to be true, then it is assumed to be false.

Its security is based on the computational intractability ( widely assumed, but not proved ) of the decisional Diffie – Hellman assumption.

His findings were mostly conditional on the Riemann hypothesis and with this assumption he found upper and lower bounds for the size of colossally abundant numbers and proved that what would come to be known as Robin's inequality ( see below ) holds for all sufficiently large values of n.

Anosov proved it under assumption.

Negation as failure is related to the closed world assumption, as it amounts to believing false every predicate that cannot be proved to be true.

Shiva proved this assumption wrong, showing that irradiating capsules with infrared light would likely never achieve ignition or gain.

Exercise as a teaching tool is deliberately omitted because of a common mistaken assumption that there exists a " correct " position.

As early as 1805, Gay-Lussac and von Humboldt showed that water is formed of two volumes of hydrogen and one volume of oxygen, and by 1811 Amedeo Avogadro had arrived at the correct interpretation of water's composition, based on what is now called Avogadro's law and the assumption of diatomic elemental molecules.

Malevich's assumption that a shifting in the attitudes of the Soviet authorities towards the modernist art movement would take place after the death of Vladimir Lenin and Leon Trotsky's fall from power, was proven correct in a couple of years, when the Stalinist regime turned against forms of abstraction, considering them a type of " bourgeois " art, that could not express social realities.

He was also correct in his assumption that neither Brazil nor Argentina paid much attention to Paraguay's interests when they formulated their policies.

In civil law jurisdictions this principle is related to the general principle of correct behavior in commercial practice — including the assumption of good faith — is a requirement for the efficacy of the whole system, so the eventual disorder is sometimes punished by the law of some systems even without any direct penalty incurred by any of the parties.

Whatever level of assumption is made, correctly calibrated inference in general requires these assumptions to be correct ; i. e., that the data-generating mechanisms really has been correctly specified.

Charles Thomas, one of Marshall's biographers, wrote that although Marshall's assumption of the presidency would have made World War II much less likely, modern hypothetical speculation on the subject was unfair to Marshall, who made the correct decision in not forcibly removing Wilson from office, even temporarily.

This contention assumes that the twins ' situations are symmetrical and interchangeable, an assumption that is not correct.

If the Zubrin / Andrews assumption were correct then d / dt ( mv < sub > 1 </ sub >) =

Apart from these there is the assumption ( hypolepsis ), which is either the hypothesis or the opinion about something ( matter or action ), and which can be correct or incorrect.

Since they are produced automatically without any rational analysis and verification ( see the modern idea of the subconscious ) of whether they are correct or not, they need to be confirmed ( epimarteresis: confirmation ), a process which must follow each assumption.

This taxonomic treatment is almost certainly erroneous, but the assumption of a close evolutionary relationship with other " higher waterbirds " – such as loons ( Gaviiformes ) and penguins ( Sphenisciformes ) – appears to be correct.

After its orbit was better determined, it was clear that Piazzi's assumption was correct and this object was not a comet but more like a small planet.

Even though this assumption is not linguistically correct as in the Iguvine Tables the god is named Fis ( i ) us or Fisovius Sancius his intrepretation, based on unknown theological documents, tallies with that of some modern scholars.

* Tautology ( rhetoric ), using different words to say the same thing, or a series of self-reinforcing statements that cannot be disproved because they depend on the assumption that they are already correct

The common assumption in photography that the pupil diameter is equal to the aperture diameter is not correct for many types of camera lens, because of the magnifying effect of lens elements in front of the aperture.

Based on the philosophical assumption of the Strong Church-Turing Universe Thesis, a mathematical criterion for evaluation of evidence has been proven, with the criterion having a resemblance to the idea of Occam's Razor that the simplest comprehensive description of the evidence is most likely correct.

The former assumption has persisted to the present day, with writers of fiction, such as Mary Renault, and the film director Oliver Stone among its proponents, as well as modern historians such as Paul Cartledge, who says: " Rumour had it — and rumour was for once surely correct — that he and Alexander had once been more than just good friends.

This assumption is not quite correct, and introduces some error.

The latter applies normally to a form of argument that is not a genuine rule of logic, where the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption.

In the assumption that the new belief is correct, some of the old ones have to be retracted in order to maintain consistency.

Jean is carrying a toddler, whose mother has died, and this leads Harman to believe that she is married ; to avoid complications, Jean does not correct this assumption.

If " Feeling the Future " is correct, it would provide evidence for psi, significantly altering the assumption of the linear nature of time, challenging the very core of modern scientific thought on the matter.