Is reincarnation false? If so, then didn't Jesus let people have a chance of reincarnating?
The doctrine of reincarnation says that when a person or animal dies, his essential essence (call it soul for convenience) is reborn, often immediately, in another body. Whether it is an improvement in situation or not depends on his karma in the life just completed. If he did evil he is reincarnated in a lesser form (such as a lower class person or an animal). If he did good he is reincarnated in a better form. He goes through this cycle of lives until he reaches a perfect state (Nirvana in Hindu theology) and loses his identity in the cosmic whole.
The Bible, on the other hand, teaches that there is no reincarnation. “And as it is appointed unto man once to die, and after that the judgement, so Christ was once offered to bear the sins of many.” (Hebrews 9:27-28)
Jesus never offered reincarnation. He did raise some people from the dead, but they were raised in their original bodies. This actually would not have been possible under classic reincarnation theory because it would involve taking the soul from the new body to put it in the old.
Nor is the final resurrection and judgement a reincarnation. The word means “becoming flesh again.” The usual understanding of the doctrine, as indicated above, is that the soul goes into another physical body to live another life or lives on earth. Paul tells the Corinthians (1 Corinthians 15:35-58) that we will not be resurrected in an earthly body. That is, we will not be re-incarnated. Instead we will be given a new, spiritual body. Since no flesh is involved in this new body, it can not, by definition, be a reincarnation.