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My question is, if a non-Jewish man marries a Jewish woman, since the marriage is invalid, are the children Jewish? Specifically, is there any source in the Mishnah which says that they are not Jewish? Are there opinions either way?

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Generally speaking, I think it's well-known that Halachah follows the mother on Jewish status. But as a Jewish Q&A site, I think this is an important question to have, along with a quality answer like Shalom's. –  Seth J Sep 17 '13 at 18:00

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This is pretty much open-and-shut.

Mishna, Kiddushin 3:13.

כל מקום שיש קידושין ואין עבירה, הוולד הולך אחר הזכר; ואיזו זו--זו כוהנת לוייה וישראלית, שנישאו לכוהן וללוי ולישראל.

...

וכל מי שאין לה לא עליו ולא על אחרים קידושין, הוולד כמוה; ואיזה זה, זה ולד שפחה ונוכרית.

Any union which is valid marriage and no prohibition, the child follows the father, namely: a female Cohen/Levite/Israelite marries a male Cohen/Levite/Israelite ...

Any union in which there would be no valid marriage between her or any Jewish man, the child follows her [the mother], namely: the child of a slave woman or non-Jewish woman.

So the mishna is spelling out that if mom is non-Jewish, the child is not Jewish. It's implying the converse fairly strongly, that if mom's Jewish so is the child; but the Gemara does quote some different opinions about it. (Rabbi Akiva, fascinatingly, is of the opinion that if dad's not Jewish but mom is, the child is a mamzer. We don't follow his opinion.)

Rambam rules (Prohibited Relations 15:4):

זה הכלל בן הבא מן העבד או מן העכו"ם או מן השפחה או מן בת עכו"ם הרי הוא כאמו ואין משגיחין על האב

This is the rule: When one parent is a slave or not Jewish, whether male or female -- the child follows the mother, and we pay no attention to [the halachic status] of the father.

So in the case of your question, it's actually precisely because no marriage is valid that we ignore the non-Jewish father completely -- leaving the child Jewish.

For theory's sake, long ago there had been an opinion that both parents must be Jewish for the child to be Jewish -- we don't follow that opinion. But Rabbi Aharon Rakeffet-Rothkoff tells the story of a female student of his whose dad was not Jewish who was constantly upset about the possibility of this minority opinion. Telling her "we don't follow that opinion" wasn't calming her fears. So he suggested that when she gets married, she'll be going to the mikvah anyhow, so she could ask a panel of three rabbis to witness the dunking and call it a conversion, just in case. Halachically it's completely not necessary, but it made this woman feel a lot better by getting rid of her doubts.

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What if a single Cohen has a child by a single Jewish woman? Maybe we would rule bdieved that the consummation effected marriage? –  yoel Sep 17 '13 at 16:57
    
@yoel Without witnesses and intent?? –  Double AA Sep 17 '13 at 18:40
    
@Yoel don't mess with the "effected marriage" mess. If the parents could theoretically have married but didn't, the child is 100% kosher (though people may still call him/her mean names). In this case, if the single woman was theoretically cohen-eligible (that's a different question), the child is a cohen too. Rabbi Akiva's opinion was that violating the laws of nida generates a mamzer, but we don't follow that opinion. –  Shalom Sep 17 '13 at 19:05
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@Shalom R Akiva agreed Niddah didn't make a Mamzer. He thought all chayvei lavin did make a mamzer though (eg, amoni and moavi, petzua daka) with the possible exception of yevama lashuk –  Double AA Sep 17 '13 at 20:20

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