What is the difference between the two adjectives ἀρχαῖος and παλαιός?
In particular, what word would fit the best to mean "history" between ἀρχαιολογία and παλαιολογία?
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The standard word for historical study was in fact archaeology. While Thucydides' "Archaeology" may be a conventional term, Josephus' Antiquitates in Greek was the Archaiologia (ἀρχαιολογία), and historical works by Cleanthes, Hieronymus the Egypytian inter alia were all titled similarly.
That said, palaios is not semantically incorrect, and Appian uses the verbal form of palaiologos once.
As far as the difference goes, they essentially both developed enough extended meanings to be nigh synonymous. Both have been used for old and ancient people and things, though archaios is generally not used for "old people," in the sense of aged to a long life, while palaios can.