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.

The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.

Paul Joseph Cohen ( April 2, 1934 — March 23, 2007 ) was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo – Fraenkel set theory, the most widely accepted axiomatization of set theory.

Cohen's proof developed the method of forcing, which is now an important tool for establishing independence results in set theory.

Ironically this act of apparent insubordination is cited by his admirers as further proof of his independence of spirit when dealing with Hitler.

The Whitehead problem on abelian groups was solved ( as an independence proof ) by Saharon Shelah.

repeat proof to show independence of X and B.

A singular independence of spirit, a breadth of mind which refused to be contracted by party formulas, a sanity which was proof against the contagion of national delirium, were equally characteristic of uncle and nephew.

The automorphisms of a Galois extension K / F are linearly independent as functions over the field K. The proof of this fact follows from a more general notion, namely, the linear independence of characters.

Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof.

Among his major accomplishments were the 1940 proof, of the Riemann hypothesis for zeta-functions of curves over finite fields, and his subsequent laying of proper foundations for algebraic geometry to support that result ( from 1942 to 1946, most intensively ).

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

The original versions of his papers contained " many technical errors of varying degree "; when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception — a claimed proof of the Continuum hypothesis.

It is important to understand that the outcome of empirical research using statistical hypothesis testing is never proof.

' In the second case, he cites an example that demonstrates ignorance of statistical principles in the lay press: ' Since no such proof is possible genetically modified food is harmless, the article in The New York Times was what is called a " bad rap " against the U. S. Department of Agriculture-a bad rap based on a junk-science belief that it's possible to prove a null hypothesis.

But for this hypothesis all proof is lacking.

In June 2004, de Branges announced he had a proof of the Riemann hypothesis ( RH ; often called the greatest unsolved problem in mathematics ) and published the 124-page proof on his website.

A confirmation of this positivity conjecture would have led to a proof of the Riemann hypothesis, as was shown by Pál Turán.

His refusal to discuss the nature of his work during this period led some to speculate that he worked for the British intelligence services in Turkey, but proof for this hypothesis is lacking.

This is a requirement on derivability, namely, the principle that in a formal system with material implication and modus ponens, if Y is provable from the hypothesis X, then there is also a proof of X → Y.

In 1965 Eugene Fama published his dissertation arguing for the random walk hypothesis, and Samuelson published a proof for a version of the efficient-market hypothesis.

These are unproven ; in 1967, Hooley published a conditional proof for the second conjecture, assuming certain cases of the Generalized Riemann hypothesis.

On the second question, no definite consensus has been formed, and the intentional discovery hypothesis lacks solid proof.

Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings ( see also the history section in the Algebraic Geometry article ).

Given the consistent failures in this research literature to disprove the null hypothesis, the burden of empirical proof is on those who argue that the children of sexual minority parents are worse off than the children of heterosexual parents.

Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests.

The Jacobian of a curve over an arbitrary field was constructed by as part of his proof of the Riemann hypothesis for curves over a finite field.

Furthermore, the authenticity of " Archaeoraptor " would not have been an essential proof for the hypothesis that birds are theropods, as this is sufficiently corroborated by other data ; paleontologist Christopher Brochu concluded in November 2001: " That birds are derived theropod dinosaurs is no longer the subject of scholarly dispute.

Malle ( 2006 ) interpreted this result not so much as proof that actors and observers explained behavior exactly the same way but as evidence that the original hypothesis was fundamentally flawed in the way it framed people's explanations of behavior — namely, as attributions to either stable dispositions or to the situation.

Thus the conjecture of Pólya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.

A good many beef-hungry settlers were accepting the death of William Lewis as proof that the warning notes were not idle threats.

I must confess that I prefer the Liberal who is personally affected, who is willing to send his own children to a mixed school as proof of his faith.

They also furnish proof that, in modern war, message sending must be monitored.

Whether you experienced the passion of desire I have, of course, no way of knowing, nor indeed have I wished with even the most fleeting fragment of a wish to know, for the fact that one constitutes by one's mere existence so to speak the proof of some sort of passion makes any speculation upon this part of one's parents' experience more immodest, more scandalizing, more deeply unwelcome than an obscenity from a stranger.

Along these lines, the particular point that sensitivity in literature leads to sensitivity in human relations would require more proof than I have seen.

In any event, the critical productivity of that time is abundant proof that if he was taking laudanum, it was never in command of him to the extent that it had been during his vagrant years.

Their policy ran counter to the traditional idea that a good fighter was usually a libertine, and that in sex affairs `` God-given passion '' was a proof of manliness.

A credulousness, a distaste for documentation, an uncritical reliance on contemporary accounts, and a proneness to assume a theory as true before adequate proof was provided were all evidences of his failure to comprehend the use of the scientific method or to evaluate the responsibilities of the historian to his reading public.

Both Alfred Harcourt and Donald Brace had written him enthusiastic praise of Elmer Gantry ( any changes could be made in proof, which was already coming from the printer ) and they had ordered 140,000 copies -- the largest first printing of any book in history.

The appointment of U Thant of Burma as the U.N.'s Acting Secretary General -- at this writing, the choice appears to be certain -- offers further proof that in politics it is more important to have no influential enemies than to have influential friends.

Also, reserve the right to demand proof of death despite the fact that you'll probably never use it.

The first is the strictly scientific, which demands concrete proof and therefore may err on the conservative side by waiting for evidence in the flesh.

The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.

The idea of the proof is this.

We shall show that the polynomials Af behave in the manner described in the first paragraph of the proof.

This completes the proof of statement ( A ).

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Both men and women of twenty-one years of age could register and vote upon presenting proof of residence and identification.

The former proof seems applicable to a statutory merger or consolidation, the latter to a contractual acquisition.

Since the change to better nutrition, he feels he can report on improvements in health, though he considers the following statements observations and not scientific proof.

so that the absence of the hymen is by no means positive proof that a girl has had sex relations.

The final proof was a small incident.

Most of them, the world over, operate on the same principle by which justice is administered in France and some other Latin countries: the customer is to be considered guilty of abysmal ignorance until proven otherwise, with the burden of proof on the customer himself.

It was unlikely that any girl as sharp as Mary Jane Brennan would believe it without proof.