For this reason Ugolino is known as the " Cannibal Count " and is often depicted gnawing at his own fingers (" eating of his own flesh ") in consternation, as in the sculpture The Gates of Hell by Auguste Rodin, in Ugolino and his Sons by Jean-Baptiste Carpeaux and in other artwork, though this may also simply refer to Ugolino's own statement in the poem that he gnawed his fingers in grief.

Ugolino's gnawing of Ruggieri's head has been interpreted as meaning that Ugolino's hatred for his enemy is so strong that he is compelled to " devour even what has no substance.

According to Dante, the prisoners were slowly starved to death and before dying Ugolino's children begged him to eat their bodies.

According to Dante, the prisoners were slowly starved to death and before dying Ugolino's children begged him to eat their bodies.

Ugolino's Rule, originally based on the Benedictine one, was amended in 1263 by Pope Urban IV to allow for the communal ownership of property, and was adopted by a growing number of monasteries across Europe.

In 1582 Vincenzo Galilei performed a setting, that he composed himself, of Ugolino's lament from Dante's Inferno.

Dōgen is known to have refuted the statement " Painted rice cakes will not satisfy hunger ".

This statement means that sutras, which are just symbols like painted rice cakes, cannot truly satisfy one's spiritual hunger.

This is partly due to a hunger strike, his decision to not appear for his opening statement ( he was self represented ), and the aforementioned witness intimidation.

In 1998, during the 68-day hunger strike of British animal-rights activist Barry Horne, Blakemore's life was threatened in a statement released by Robin Webb of the Animal Liberation Press Office on behalf of the Animal Rights Militia.

The conductor did recall having priests as passengers and this satisfied police, although the conductor also pointed out that in heavily Catholic Fall River there were priests riding on almost every trip the streetcar made, so Morse's statement really proved nothing.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

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.

In a sense it is more of a research program proposal: part of the research is to find a proper formal statement that is physically reasonable and that can be proved to be true or false ( and that is sufficiently general to be interesting ).

The following more general statement will be proved.

If a definite statement is believed plausible by some mathematicians but has been neither proved nor disproved, it is called a conjecture, as opposed to an ultimate goal: a theorem that has been proved.

For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of " A or not A ".

The statement made in the same source, that Linus suffered martyrdom, cannot be proved and is improbable.

According to the theorem, within every sufficiently powerful logical system ( such as Principia ), there exists a statement G that essentially reads, " The statement G cannot be proved.

In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module.

Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent ( both a statement and its denial can be derived from its axioms ) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.

* A converse of a theorem is a statement formed by interchanging what is given in a theorem and what is to be proved.

In the converse, the given ( that two sides are equal ) and what is to be proved ( that two angles are equal ) are swapped, so the converse is the statement that if two angles of a triangle are equal then two sides are equal.

Using an equivalent form of the paradox which reduces the length of the week to just two days, he proved that although self-reference is not illegitimate in all circumstances, it is in this case because the statement is self-contradictory.

'" The case was dismissed in July 2007 when the judge ruled that Condit had not proved the statement was false, or that the paper had published it with malice.

Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom.

Most of the social and economic goals announced at the eighth Congress had been reached ; however, the absence of a definitive statement on further efforts to improve the working and living conditions of the population proved to be a source of concern.

On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.

We use the stronger statement that every odd ( antipode-preserving ) mapping h: S < sup > n-1 </ sup > → S < sup > n-1 </ sup > has odd degree.

For clarity, this statement asserts that there is at least one person who can be fooled at all times, which is stronger than asserting that at all times at least one person exists who can be fooled.

If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.

However, this is not the stronger sense of completeness desired for Principia Mathematica, since a given system of axioms ( such as those of Principia Mathematica ) may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.

* The pointwise limit of a sequence of measurable functions is measurable ; note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.

Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. showed that it is unprovable in Peano arithmetic ( but it can be proven in stronger systems, such as second order arithmetic ).

So, for example, " a or b " is a stronger statement than " if not a, then b ", whereas these are classically interchangeable.

Thus, if the permanent can be computed in polynomial time by any method, then FP = # P, which is an even stronger statement than P = NP.

Moreover, this stronger statement can be proved as well to be accomplished by means of translations only.

Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture.

No stronger proof of his pre-eminence could be given than the oft-quoted saying of Henry Hardinge, 1st Viscount Hardinge that one of John Wilson Croker's ablest speeches was demolished by the simple statement of Lord Althorp that he " had collected some figures which entirely refuted it, but had lost them.

The modern statement of the theorem is both more general and stronger than the version for countable signatures stated in the introduction.

The order ord < sub > n </ sub > a also divides λ ( n ), a value of the Carmichael function, which is an even stronger statement than the divisibility of φ ( n ).

A common statement of the lemma ( as given below ) makes the stronger assumption that the theory can represent all computable functions.

A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it " teleport " from one place to another — it can only move by a continuous flow.

An even stronger statement is Gabber's theorem, which states that Char ( M ) is a co-isotropic subvariety of for the natural symplectic form.

When ƒ is in addition continuous, and both ƒ and ƒ ^ decay sufficiently fast at infinity, then one can " invert " the domain back to R < sup > d </ sup > and make a stronger statement.

The first one is called " the strong AI hypothesis " and the second is " the weak AI hypothesis " because the first one makes the stronger statement: it assumes something special has happened to the machine that goes beyond all its abilities that we can test.

In January 2011, during a White House meeting with the President of France Nicolas Sarkozy, Obama declared: " We don't have a stronger friend and stronger ally than Nicolas Sarkozy, and the French people ", a statement which triggered outcry in the United Kingdom.