In medical testing, and more generally in binary classification, a false positive is an error in data reporting in which a test result improperly indicates presence of a condition, such as a disease (the result is positive), when in reality it is not present, while a false negative is an error in which a test result improperly indicates no presence of a condition (the result is negative), when in reality it is present.
https://en.wikipedia.org/wiki/False_positives_and_false_negatives
Also, it looks like the Philadelphia population is just over 1.5 million*, so if we say 1.5 million have not yet had the virus, then 1 in 500 of that is 3,000, not 10,000, so:
Of those who have antibodies, the "100%" claim suggests that anyone with antibodies will correctly test positive with zero false negatives.
Of those who do not have antibodies then the "99.8%" claim suggests that 99.8% will correctly test negative, and that .2% will falsely test positive.
Which, in Philadelphia, would mean that 3,000 people would be told they are not at risk of infection, when they actually are. (assuming we've nailed down, per other news, that the risk of reinfection is very low)
That is not an error on the safe side outcome. It would suck to be one of those 3,000 people if they changed their behaviour and subsequently caught it.
*Unless you mean the larger Metropolitan Statistical Area of Philadelphia-Camden-Wilmington, which has about 6 million:
https://en.wikipedia.org/wiki/List_of_metropolitan_statistical_areas