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proof and independence
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.
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.
A proof of the independence of the continuum hypothesis.
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.

proof and result
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 ).
This key result opened the way for a proof of the Weil conjectures, ultimately completed by his student Pierre Deligne.
The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the " wrong " order.
Probability of loss is generally an empirical exercise, while cost has more to do with the ability of a reasonable person in possession of a copy of the insurance policy and a proof of loss associated with a claim presented under that policy to make a reasonably definite and objective evaluation of the amount of the loss recoverable as a result of the claim.
This result, known as Tarski's undefinability theorem, was discovered independently by Gödel ( when he was working on the proof of the incompleteness theorem ) and by Alfred Tarski.
The Essenes have gained fame in modern times as a result of the discovery of an extensive group of religious documents known as the Dead Sea Scrolls, which are commonly believed to be Essenes ' library — although there is no proof that the Essenes wrote them.
Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals.
Upon a proof of res ipsa loquitur, the plaintiff need only establish the remaining two elements of negligence — namely, that the plaintiff suffered harm, of which the incident result was the legal cause.
Finally, we provide a proof of the related result, rk ( A ) =
Maher's proof appears to contradict the result of the Bayesian argument, which was that the observation of a non-black non-raven provides much less evidence than the observation of a black raven.
These cases demonstrate a paradox not in the sense that they demonstrate a logical contradiction, but in the sense that they demonstrate a counter-intuitive result that is provably true: the situations " there is a guest to every room " and " no more guests can be accommodated " are not equivalent when there are infinitely many rooms ( an analogous situation is presented in Cantor's diagonal proof ).
This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological.
A major result of complexity theory is that PSPACE can be characterized as all the languages recognizable by a particular interactive proof system, the one defining the class IP.
However, the result only holds under the restrictive assumptions necessary for the proof ( markets exist for all possible goods so there are no externalities, all markets are in full equilibrium, markets are perfectly competitive, transaction costs are negligible, and market participants have perfect information ).
Within this element lies an implicit reversal of the onus of proof: under the precautionary principle it is the responsibility of an activity proponent to establish that the proposed activity will not ( or is very unlikely to ) result in significant harm.
These two theorems are very different from each other ; the first one has a very simple proof and is very counterintuitive, while the proof of the second one is very technical but the result is not at all surprising.
The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from the above result.
For a proof of this result, see the corresponding section on the Hilbert's basis theorem page.
His treatise upon baptism is the work of a scholar, and places before the reader that exhaustive proof that immersion is the one and only action commanded by the Saviour which can only be reached as the result of complete classical research.
This result, known as Tarski's undefinability theorem, was discovered independently by Gödel ( when he was working on the proof of the incompleteness theorem ) and by Alfred Tarski.

proof and also
They also furnish proof that, in modern war, message sending must be monitored.
Schwab also declared there is no proof of Weinstein's entering a conspiracy to use the U.S. mails to defraud, to which federal prosecutor A. Lawrence Burbank replied:
It also brought home proof of something a casual observer might have missed: that more than half of the U.S. Negroes live outside the southeastern states.
A grant application to build a proof of concept prototype was submitted in March 1939 to the Agronomy department which was also interested in speeding up computation for economic and research analysis.
That was so because it not only was proof of excessive pride, but also resulted in violent acts by or to those involved.
A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.
In calculus, this picture also gives a geometric proof of the derivative if one sets and interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, where the coefficient of the linear term ( in ) is the area of the n faces, each of dimension
In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it.
Banach's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.
Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.
The engraving may also be the first visual proof of cricket being played in the United States.
Concepts such as infinite proof trees or infinite derivation trees have also been studied, e. g. infinitary logic.
To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.
First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim – Skolem theorem and the compactness theorem.
Cantor also published an erroneous " proof " of the inconsistency of infinitesimals.
From these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz's law, the identity of indiscernibles: two or more objects are identical ( are one and the same ) if they have all their properties in common, and so, there would only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness.
" Hello world " is also used by computer hackers as a proof of concept that arbitrary code can be executed through an exploit where the system designers did not intend code to be executed — for example, on Sony's PlayStation Portable.
A writ of habeas corpus, also known as the Great Writ, is a summons with the force of a court order ; it is addressed to the custodian ( a prison official for example ) and demands that a prisoner be taken before the court, and that the custodian present proof of authority, allowing the court to determine whether the custodian has lawful authority to detain the person.
It is also agreed that " no direct or categorical and stringent proof of the dogma can be brought forward from Scripture ".
He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
Other authors, like Oruch, have also claimed that the modern customs of Saint Valentine's Day originate from Lupercalia customs, again without proof.
Beyond programming languages, the lambda calculus also has many applications in proof theory.
Modern tests and contemporary accounts agree therefore that well-made plate armour could protect against longbows, however there are a number of caveats to this point ; not all plate armour was well-made or well looked after, and there were also weak points in the eye and air holes and joints where arrows could penetrate, meaning that even if the armour was proof against nearly all arrows, being shot at by thousands of longbowmen would have been an uncomfortable experience, physically and mentally.
Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic.

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