In the Catholic Church ( both the Western and Eastern Catholic Churches ) the act of canonization is reserved to the Holy See and occurs at the conclusion of a long process requiring extensive proof that the person proposed for canonization lived and died in such an exemplary and holy way that he or she is worthy to be recognized as a saint.

Therefore, Socrates opposed the Sophists and their teaching of rhetoric as art and as emotional oratory requiring neither logic nor proof.

Badges, rings, decoding devices and other radio premiums offered on these adventure shows were often allied with a sponsor's product, requiring the young listeners to mail in a box top from a breakfast cereal or other proof of purchase.

This is perhaps the simplest known proof, requiring the least mathematical background.

Early laws simply prohibited driving while intoxicated, requiring proof of a state of intoxication with no specific definition of what level of inebriation qualified.

Jim Crow legislation related to voting would quietly disenfranchise the Southern African American by requiring of prospective voters proof of land ownership or literacy tests at poll stations.

John Schlight, in his A War Too Long, said of the PAVN's logistical apparatus :" This sustained effort, requiring the full-time activities of tens of thousands of soldiers, who might otherwise have been fighting in South Vietnam, seems proof positive that the bombing of the Ho Chi Minh Trail had disrupted the North Vietnamese war effort.

That let Peirce frame scientific inquiry not only as a special kind of inquiry in a broader spectrum, but also, like inquiry generally, as based on actual doubts, not mere verbal doubts ( such as hyperbolic doubt ), which he held to be fruitless, and it let him also frame it, by the same stroke, as requiring that proof rest on propositions free from actual doubt, rather than on ultimate and absolutely indubitable propositions.

In 2006, the then Australian attorney-general Philip Ruddock had rejected calls by two reports — from a Senate committee and the Australian Law Reform Commission — to limit the sedition provisions in the Anti-Terrorism Act 2005 by requiring proof of intention to cause disaffection or violence.

By 1800 the other provinces of British North America had effectively limited slavery through court decisions requiring the strictest proof of ownership, which was rarely available.

A presumption of constitutionality shifts the burden of proof from the government to the citizen, requiring them to prove that a statute is unconstitutional.

One of the main challenges to renewal experienced with some of these policies is requiring proof of insurability.

The fanciful design and manufacturer's logo commonly displayed on the Ace of Spades began under the reign of James I of England, who passed a law requiring an insignia on that card as proof of payment of a tax on local manufacture of cards.

To do so would, in the words of Commissioner Morling, involve "... ( a ) fundamental error of reversing the onus of proof and requiring Mrs Chamberlain to prove her innocence.

To examine the evidence to see whether it has been proved that a dingo took Azaria would be to make the fundamental error of reversing the onus of proof and requiring Mrs Chamberlain to prove her innocence.

In addition, several states have passed legislation requiring proof of identification and a written record of all pseudoephdrine sales.

Writing professionals hold that, “ In a rhetorical argument, a fact is a claim that an audience will accept as being true without requiring proof .” ( Kantz, 76 ) Facts can be thought of merely as claims.

Other amendments included requiring the Director of Public Prosecutions to make a statement that a prosecution would be impossible before each individual control order could be issued, to require a judge to authorise each control order, requiring a review of the legislation by Privy Councillors and restoring the " normal " burden of proof (" beyond a reasonable doubt "), rather than the weaker " balance of probabilities ".

Applicants should show the reason for requiring a PPS Number, and provide Photographic ID and proof of address.

** Non-interactive zero-knowledge proof, a common random string shared between the prover and the verifier is enough to achieve computational zero-knowledge without requiring interaction

Some places requiring proof of age will not accept some of the cards available.

Local officials, however, were already requiring betrothed couples to prove they were worthy to marry by presenting proof of Aryan ancestry.

15 % of the exam is " multiple choice ", 15 % " fill in the gaps ", the rest requiring detailed explanations and proof.

The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true.

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.

In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.

There are several results in category theory which invoke the axiom of choice for their proof.

The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma, one formulation of the axiom of choice.

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.

The following construction of the Vitali set shows one way that the axiom of choice can be used in a proof by transfinite induction:

The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

* An axiom or postulate is a statement that is accepted without proof and regarded as fundamental to a subject.

If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps.

To verify a formal proof when the set of axioms is infinite, it must be possible to determine whether a statement that is claimed to be an axiom is actually an axiom.

The difficulty is that there is no mechanical way to decide, given a statement about the natural numbers, whether it is an axiom of this theory, and thus there is no effective way to verify a formal proof in this theory.

He was able to develop most of classical calculus, while using neither the axiom of choice nor proof by contradiction, and avoiding Georg Cantor's infinite sets.

When proving basic results about the natural numbers in elementary number theory though, the proof may very well hinge on the remark that any natural number has a successor ( which should then in itself be proved or taken as an axiom, see Peano's axioms ).

In order to establish his proof, Reinhold stated an axiom that could not possibly be doubted.

Unlike the generalized solution to Tarski's circle-squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out " physically "; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.

* Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference

Thoralf Skolem ( 1920 ) gave a correct proof using formulas in what would later be called Skolem normal form and relying on the axiom of choice:

While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma, which is strictly weaker ( the proof given below, however, assumes trichotomy, i. e., that all cardinal numbers are comparable, a statement which is also equivalent to the axiom of choice ).