Actually, he has it right. Our modern, western notion of a calendar is marred by the fact that the Romans had no concept of zero until the conquest of Spain and the ensuing interaction with the moorish people who lived there. Thus, we start counting dates with 1, not zero. Therefore, the '60's is the decade beginning immediately after the end of year xx60 but a person "in their 60's" has completed 59 years of life and not 10 more.
In our Christian era calendars you do not find a year zero. To our modern, mathematically educated minds that would have been the year before Jesus of Nazareth was 12 months old.
Of course, our calendars, while allegedly based on the birth date of this man Jesus, are flawed by many other issues. Among these are:
1) We don't actually have agreement about the precise year of Jesus' birth.
2) The 25-December customary date is a fabrication. Jesus was most likely born in the spring based on accounts of what was happening at the time.
3) Our calendar system has been changed a few times over the past two millennia.