Maximian's appointment is unusual in that it was impossible for Diocletian to have been present to witness the event.

Galerius and Constantius would become Augusti after Diocletian and Maximian's departure.

There are suggestions in the Panegyrici Latini and Lactantius ' account that Diocletian arranged plans for his and Maximian's future retirement of power in Rome.

In 293, feeling more focus was needed on both civic and military problems, Diocletian, with Maximian's consent, expanded the imperial college by appointing two Caesars ( one responsible to each Augustus ) — Galerius and Constantius Chlorus.

He shared the latter title with his co-emperor and superior, Diocletian, whose political brain complemented Maximian's military brawn.

Maximian's swift appointment by Diocletian as Caesar is taken by the writer Stephen Williams and historian Timothy Barnes to mean that the two men were longterm allies, that their respective roles were pre-agreed and that Maximian had probably supported Diocletian during his campaign against Carinus ( r. 283 – 285 ) but there is no direct evidence for this.

Diocletian, Maximian's senior colleague and Augustus in the east

Maximian's campaigns were not proceeding as smoothly.

He was Maximian's praetorian prefect in Gaul, and the husband to Maximian's daughter, Theodora.

Maximian's son Maxentius, and Constantius ' son Constantine would then become Caesars.

He heard of Maximian's third claim to the throne, his forced suicide, his damnatio memoriae.

Constantine forced Maximian's suicide in 310.

* Constantius Chlorus divorces his wife ( or concubine ) Helena, mother of Constantine I, and marries emperor Maximian's stepdaughter, Theodora.

He married Theodora, Maximian's daughter.

Maximian's subordinate, Constantius, campaigned against Carausius ' successor, Allectus, while Maximian held the Rhine frontier.

In April 307, he attempted to depose his son, but failed and fled to the court of Constantius ' successor, Constantine ( who was both Maximian's step-grandson and also his son-in-law ), in Trier.

During Constantine's war with Maxentius, Maximian's image was purged from all public places.

However, after Constantine ousted and killed Maxentius, Maximian's image was rehabilitated, and he was deified.

Theodora, the wife of Constantius Chlorus, is often called Maximian's stepdaughter by ancient sources, leading to claims by Otto Seeck and Ernest Stein that she was born from an earlier marriage between Eutropia and Afranius Hannibalianus .< ref > Aurelius Victor, de Caesaribus 39. 25 ; Eutropius, Breviaria 9. 22 ; Jerome, Chronicle 225 < sup > g </ sup >; Epitome de Caesaribus 39. 2, 40. 12, quoted in Barnes, New Empire, 33 ; Barnes, New Empire, 33 .</ ref > Barnes challenges this view, saying that all " stepdaughter " sources derive their information from the partially unreliable work of history Kaisergeschichte, while other, more reliable sources, refer to her as Maximian's natural daughter .< ref > Origo Constantini 2 ; Philostorgius, Historia Ecclesiastica 2. 16 < sup > a </ sup >, quoted in Barnes, New Empire, 33.

The perceived creation of a " discount " on individual menu items in exchange for the purchase of a " meal " is also consistent with the loyalty marketing school of thought.

Many of the masses felt resentment towards the increasingly corrupt government ; their loyalty to the clergy, who were seen as more concerned with the fate of the populace, remained consistent or increased.

On 24 September James II visited the Duke at Badminton, and expressed his satisfaction at his consistent loyalty.

Tien made clear that he was personally infuriated by Chow's unfriendly act despite the Liberal Party's loyalty and consistent support for the rail company.

ACES created a corporate culture of service and punctuality that would earn it the recognition and loyalty of its customers, resulting in consistent increases in its market share and revenues in the domestic market, which was dominated for decades by Avianca.

Royce himself declared the “ successive expressions ” of the philosophy of loyalty “ form a consistent body of ethical as well as religious opinion and teaching, verifiable, in its main outlines, in terms of human experience, and capable of furnishing a foundation for a defensible form of metaphysical idealism .” ( The Problem of Christianity, Vol.

The Truman Order — based on the findings of an interdepartmental committee established in 1946 — was superseded by President Dwight D. Eisenhower ’ s issuance of Executive Order 10450 in April 1953, which provided that “ he appointment of each civilian officer or employee in any department or agency of the Government shall be made subject to an investigation ,” and made each agency head responsible for ensuring that “ the employment and retention in employment of any civilian officer or employee within the department or agency is clearly consistent with the interests of the national security .” While abolishing the Truman Order program, including the Loyalty Review Boards within the Civil Service Commission, the new Order also made clear that “ the interests of national security require that all persons privileged to be employed in the departments and agencies of the Government, shall be reliable, trustworthy, of good conduct and character, and of complete and unswerving loyalty to the United States .”

Although toned down slightly for the younger-child friendly series, as were many of the characters, his cartoon portrayal is generally consistent with his mini-comic portrayal, although in some early episodes the show's writers added extra dimensions to his character in that despite his loyalty, he clearly resents being bossed around by Skeletor and secretly desires to someday overthrow his master.

Ranulf de Blondeville, 6th Earl of Chester and 1st Earl of Lincoln ( 1172 – 1232 ), known in some references as the 4th Earl of Chester ( in the second lineage of the title after the original family line was broken after the 2nd Earl ), was one of the " old school " of Anglo-Norman barons whose loyalty to the Angevin dynasty was consistent but contingent on the receipt of lucrative favours.

Diocletian and the Tetrarchs had no consistent plan for frontier advancement, and records of raids and forts built across the frontier are likely to indicate only temporary claims.

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 ).

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.

The 1976 definition of the astronomical unit was incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and " vigorous debate " ensued until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = 149597870700 meters.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates.

Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain sense: there is an algorithm that, for every proposition, can be shown either true or false.

In any event, Kurt Gödel in 1930 – 31 proved that while the logic of much of PM, now known as first-order logic, is complete, Peano arithmetic is necessarily incomplete if it is consistent.

In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were.

) A formal system is consistent if for all formulas φ of the system, the formulas φ and ¬ φ ( the negation of φ ) are not both theorems of the system ( that is, they cannot be both proved with the rules of the system ).

For each consistent formal theory T having the required small amount of number theory, the corresponding Gödel sentence G asserts: " G cannot be proved within the theory T ".

This means that if the theory T is consistent then G cannot be proved within it, and so the theory T is incomplete.

The sightings proved consistent with accounts of previous phantom cat sightings in the eastern United States.

" At what is now known as Oil Creek, Butler County resident William Smith and Edwin Drake first proved that oil could be tapped from underground for consistent supply.

It was proved consistent by defining point to mean a pair of antipodal points on a fixed sphere and line to mean a great circle on the sphere.

What Hilbert wanted to do was prove a logical system S was consistent, based on principles P that only made up a small part of S. But Gödel proved that the principles P could not even prove P to be consistent, let alone S!

It showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic on the ( infinite ) set of natural numbers – a statement that formally expresses its own unprovability, which he then proved equivalent to the claim of consistency of the theory ; so that ( assuming the consistency as true ), the system is not powerful enough for proving its own consistency, let alone that a simpler system could do the job.

As with all large cardinals, none of these varieties of Mahlo cardinals can be proved to exist by ZFC ( assuming ZFC is consistent ).

The consistency strength is Σ-indescribable cardinals is below that of Π-indescribable, but for m > 1 it is consistent with ZFC that the least Σ-indescribable exists and is above the least Π-indescribable cardinal ( this is proved from consistency of ZFC with Π-indescribable cardinal and a Σ-indescribable cardinal above it ).

Furthermore, it is consistent with ZF that every aleph bigger than is singular ( a result proved by Moti Gitik ).

For example, Gorman ( 1981 ) proved that a system of Engel curves must have a matrix of coefficients with rank three ( or less ) in order to be consistent with utility maximization.

In 1929, Presburger proved that Peano arithmetic without multiplication was consistent, complete, and decidable.

Criticisms include that the TAT is unscientific because it cannot be proved to be valid ( that it actually measures what it claims to measure ), or reliable ( that it gives consistent results over time ).

( The theories that have been proved consistent by these methods are quite strong, and include most " ordinary " mathematics.