Every moment begins "a" new millennium. January 1, 2000 certainly begins "a" millenniumm; it also begins "the" millennium of the 2000s. It doesn't begin the 3rd Millennium A.D. which starts on January 1, 2001. There is no contradiction except when people try to identify the millennium of the 2000s with the 3rd millennium A.D. The two overlap in 999 years, but identifying the two exactly is mathematically impossible.

My work will require me be on call on New Years' eve. I hope there will be catters to chat with while I wait around.

T.