An extract on #artofvisuals
As discussed above, in ZFC, the axiom of choice is able to provide "nonconstructive proofs" in which the existence of an object is proved although no explicit example is constructed. ZFC, however, is still formalized in classical logic. The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed. The status of the axiom of choice varies between different varieties of constructive mathematics.
In Martin-Lf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem. Errett Bishop argued that the axiom of choice was constructively acceptable, saying
A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.
In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of excluded middle (unlike in Martin-Lf type theory, where it does not). Thus the axiom of choice is not generally available in constructive set theory. A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.
Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory. Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
In 447, Attila again rode south into the Eastern Roman Empire through Moesia. The Roman army, under Gothic magister militum Arnegisclus, met him in the Battle of the Utus and was defeated, though not without inflicting heavy losses. The Huns were left unopposed and rampaged through the Balkans as far as Thermopylae.
Constantinople itself was saved by the Isaurian troops of magister militum per Orientem Zeno and protected by the intervention of prefect Constantinus, who organized the reconstruction of the walls that had been previously damaged by earthquakes and, in some places, to construct a new line of fortification in front of the old. An account of this invasion survives:
The barbarian nation of the Huns, which was in Thrace, became so great that more than a hundred cities were captured and Constantinople almost came into danger and most men fled from it. ... And there were so many murders and blood-lettings that the dead could not be numbered. Ay, for they took captive the churches and monasteries and slew the monks and maidens in great numbers.