Jews believe in reincarnation? I do not thing so.
Or do you mean even Jews believe in some sort of dealing of people who were not perfectly good?
As R_C has already showed the concept of Purgatory comes from many passages in the Bible.
The point is: only what is perfectly pure can go into heaven. Most of us are often stained by sin, even if forgiven by God, an still often have earthly attachments.
Still we desire God and want to be with him and try to live a good life. We do not deserve Hell. Yet we cannot go into Heaven as we are. So God PURIFIES US. ‘Purgatory’ means literally ‘place of purification/purging’, from Latin ‘purgare’ (to purge).
Whether it is a place or just a process, whether is lasts for a certain time or it is instantaneous, it’s still debated.
The Church teaches there is such thing as purgatory, and if we are not perfectly pure we must go through it. The details are not part of the ‘dogma’, however.
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I think this is a very radical position for a Jew. It’s like a Christian believing in reicarnation, essentially, not part of the orthodox faith.
If you read the Jewish Encyclopedia on the subjects:
jewishencyclopedia.com/articles/14479-transmigration-of-souls
it becomes clear that Judaism does NOT believe in ‘Transmigration of the soul’ and reincarnation, except, perhaps, a few people who are not orthodox.
I think that would be news to most Jews as well!
This is not what the bible or Jesus taught however, but also a very uncommon position.
Jesus talks several times about Hell and he does not speak of it as a ‘repeat on Earth’…