Mishnah 13 in the third chapter of Kiddushin addresses a fundamental question in the laws of lineage: is there a way for mamzerim to be purified, meaning that their descendants would no longer be considered mamzerim? Two Tannaim disagree about this.
The View of Rabbi Tarfon:
Rabbi Tarfon says: "Yecholin mamzerim litaher" - there is a way by which a mamzer can purify his descendants, so that they will not be mamzerim. How so? The Mishnah describes the process in three stages:
"Mamzer she'nasa shifchah" - the mamzer marries a Canaanite maidservant.
"Ha'vlad eved" - the child does not follow the father. As explained in the previous Mishnah, in the case of a maidservant or a non-Jew, lineage is determined by the mother alone, and therefore the son is a slave.
"Shichriro - nimtza ha'ben ben chorin" - once the child is freed, whether a son or a daughter, he becomes a free person and a fully valid Jew.
The principle behind this: the mamzerus comes from the father's Jewish side, yet here the lineage does not follow the father. Even though the father is a mamzer, his mamzerus is not transmitted to the child, since the child is not Jewish at all but a slave or a maidservant. Therefore, when he is freed and brought into the Jewish people, he enters in purity. This is the way in which mamzerim can be purified.
The View of Rabbi Eliezer:
Rabbi Eliezer disagrees and says: "Harei zeh eved mamzer" - the child is both a slave and a mamzer. In his opinion, the status of mamzerus does continue, and it can apply even to one who is a slave or a maidservant. The mamzerus leaves its mark, at least in the case of a slave and a maidservant, since they are not entirely disconnected from the Jewish people.
In summary: in this Mishnah the Tannaim disagree about whether there is a path of purification for mamzerim. According to Rabbi Tarfon, when a mamzer marries a maidservant, their child is a slave who follows his mother, and once he is freed he is a free person of valid lineage, since the mamzerus on the father's side is not transmitted to him. According to Rabbi Eliezer, the child is a slave who is also a mamzer, since the mamzerus takes effect and continues even in him.