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Actually, the Bible makes no such claim. That age is the spawn of man -- not of God. Specifically, it was this chap
who, ca 1658 wrote this jewel of logic:
For as much as our Christian epoch falls many ages after the beginning of the world, and the number of years before that backward is not only more troublesome, but (unless greater care be taken) more lyable to errour; also it hath pleased our modern chronologers, to adde to that generally received hypothesis (which asserted the Julian years, with their three cycles by a certain mathematical prolepsis, to have run down to the very beginning of the world) an artificial epoch, framed out of three cycles multiplied in themselves; for the Solar Cicle being multiplied by the Lunar, or the number of 28 by 19, produces the great Paschal Cycle of 532 years, and that again multiplied by fifteen, the number of the indiction, there arises the period of 7980 years, which was first (if I mistake not) observed by Robert Lotharing, Bishop of Hereford, in our island of Britain, and 500 years after by Joseph Scaliger fitted for chronological uses, and called by the name of the Julian Period, because it conteined a cycle of so many Julian years. Now if the series of the three minor cicles be from this present year extended backward unto precedent times, the 4713 years before the beginning of our Christian account will be found to be that year into which the first year of the indiction, the first of the Lunar Cicle, and the first of the Solar will fall. Having placed there fore the heads of this period in the kalends of January in that proleptick year, the first of our Christian vulgar account must be reckoned the 4714 of the Julian Period, which, being divided by 15. 19. 28. will present us with the 4 Roman indiction, the 2 Lunar Cycle, and the 10 Solar, which are the principal characters of that year.We find moreover that the year of our fore-fathers, and the years of the ancient Egyptians and Hebrews were of the same quantity with the Julian, consisting of twelve equal moneths, every of them conteining 30 days, (for it cannot be proved that the Hebrews did use lunary moneths before the Babylonian Captivity) adjoying to the end of the twelfth moneth, the addition of five dayes, and every four year six. And I have observed by the continued succession of these years, as they are delivered in holy writ, that the end of the great Nebuchadnezars and the beginning of Evilmerodachs (his sons) reign, fell out in the 3442 year of the world, but by collation of Chaldean history and the astronomical cannon, it fell out in the 186 year c Nabonasar, and, as by certain connexion, it must follow in the 562 year before the Christian account, and of the Julian Period, the 4152. and from thence I gathered the creation of the world did fall out upon the 710 year of the Julian Period, by placing its beginning in autumn: but for as much as the first day of the world began with the evening of the first day of the week, I have observed that the Sunday, which in the year 710 aforesaid came nearest the Autumnal Æquinox, by astronomical tables (notwithstanding the stay of the sun in the dayes of Joshua, and the going back of it in the dayes c Ezekiah) happened upon the 23 day of the Julian October; from thence concluded that from the evening preceding that first day of the Julian year, both the first day of the creation and the first motion of time are to be deduced.
— J. Ussher, The Annals of the World iv (1658)
Even so, Ussher himself claimed that Genesis was inadequate to his his task, so he depended on five other "authorities" in determining his ca 6,000-year age of Creation...