According to Orthodox Jewish tradition, in addition to the possibility that one becomes a “naturalized Jew” by means of conversion to Judaism, the religion one is born into is determined by the religious status of one’s mother, religious lineage being defined as matrilineal. The biblical basis for this assumption is Deuteronomy 7:3-4 :
Neither shalt thou make marriages with them (the non-Jewish nations that will be encountered when the Jewish people enter the land of Israel): thy daughter thou shalt not give unto his son, nor his daughter shalt thou take unto thy son. For he (the non-Jewish father) will turn away thy son from following Me, that they may serve other gods…
The Oral Tradition in Tractate Kiddushin 68b interprets the implication of these verses:
How do we know that her children bears her (religious) status? — R. Yochanan said on the authority of R. Shimeon b. Yochai, Because Scripture saith, “For he will turn away thy son from following me”:thy son byan Israelite woman is called thy son, but thy son by a non-Jewish woman is not called thy son.Ravina said: This proves that thy daughter's son by a non-Jewish man is called thy son.
The subtle inference drawn by the Talmud from the biblical verses is that since despite the fact that Deuteronomy 7:3 mentions both possibilities, i.e., a Jewish man marrying a non-Jewish woman, as well as a Jewish woman marrying a non-Jewish man, the specific fear delineated in 7:4 regards exclusively concerns itself with what will happen to a son who will come under the influence of a non-Jewish father. Consequently, when the mother is Jewish, the child is “your son”, but, by implication, not when the mother is not Jewish. The Codes accept this definition, as in Shulchan Aruch, Even HaEzer 4:5.
Consequently, with respect to the specific question whether the child of a Jewish mother is considered Jewish, the answer according to Orthodoxy is yes. As to whether that individual’s children are Jewish or not, that would then depend upon the individual’s gender. If the person is a woman, then just as she is Jewish, her children are similarly Jewish. On the other hand, if we are speaking about a man, then it would depend upon with whom, i.e., a Jewish or non-Jewish woman, he has children.
However, even if an individual is “technically” Jewish by virtue of his biological origins, whether or not he views himself as Jewish, let alone if he lives in accordance with the dictates of a Jewish lifestyle, is certainly within his own purview. Ideally, being Jewish is not just a matter of biology and/or conversion, but also includes the means by which an individual chooses to live his or her life.
For nearly 2,000 years, Jewish status has been passed matrilineally - through the mother. The precise theory behind this position or historical source for this rule may be debatable. But as a handy explanation for why, I would cite the Talmudic view that "a fetus is considered a part of the mother's body." This means that a baby gestated by a Jewish woman is considered to be constituted as a Jewish body already. (And obviously a person gestated by a non-Jewish woman would have gentile status.)
So as a theoretical matter, if you could demonstrate that your birth mother was Jewish, then yes, by traditional Jewish law you'd be Jewish too. The questioner has not said whether he or she is a he or a she. So whether your own children are born Jewish would follow the status of their birth mother - whether that's the questioner herself or the questioner's wife.
I will leave it to more liberal Jews to explain how this person might be treated under patrilineal views. My inclination as a Conservative Jew would be to see the questioner's Jewish birth as something like satisfying a biological pre-condition, but not itself sufficient reason to be seen as fully Jewish. That is, if you never knew your birth mother, and you had no Jewish upbringing or education, and were raised by a non-Jewish father, then you probably have only the barest possible sense of a Jewish identity.
As a matter of Halakha, Jewish law, no, you don't need to convert to be a Jew. But a biological qualification as a Jew can only get you in the door of Jewish community. We'll be glad to welcome you home. But once you walk in the door, we want to know: how are you going to build a full Jewish identity? What are you going to study about the faith and culture of your maternal ancestors? What practices will you take on to shape a Jewish life?
If you mother is Jewish then you are Jewish and it makes you children Jewish according to the Halacha (Jewish law). However, being Jewish is more than biological. It is also being knowledgible and observant. Therefore I strongly recommend that you contact a rabbi about how best for you and children to claim your Jewish identity and to strengthen your relationship to Judaism and the Jewish people.
For example in 1983 the Central Conference of American Rabbis passed a resolution on the status of chilren of mixed marriages
Its conclusion is as follows:
The Central Conference of American Rabbis declares that the child of one Jewish parent is under the presumption of Jewish descent. This presumption of the Jewish status of the offspring of any mixed marriage is to be established through appropriate and timely public and formal acts of identification with the Jewish faith and people. The performance of these mitzvot serves to commit those who participate in them, both parent and child, to Jewish life.
Depending on circumstances,1mitzvot leading toward a positive and exclusive Jewish identity will include entry into the covenant, acquisition of a Hebrew name, Torah study, Bar/Bat Mitzvah, and Kabbalat Torah (Confirmation).2 For those beyond childhood claiming Jewish identity, other public acts or declarations may be added or substituted after consultation with their rabbi
This implies that it is important to become knowledgible and observance are key elements is being Jewish and that biology is only one factor in an open society
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