Actually, the “logic” is that either people are bodies and souls, who die and have a final fate, or they don’t actually count as having died – they would just be spirits without any native body, who have “death” periods as part of their lifecycle, and who possess various sorts of biological creatures.
Also, it would mean that you never actually get to heaven, the material world logically would have no end, etc.
Reincarnation is basically a way to deny normal beings the chance to achieve any sort of eternal happiness and perfect virtue. St. Irenaeus goes into detail about it; but yeah, it’s basically one more serving of Gnostic “Matter is yucky and would contaminate God, so how dare anyone get into personal contact with the perfectness of God.” It also denies people any kind of permanent moral decision; any differentiation from other individual beings; any specialness for human beings that sets them above the animals, plants, or even rocks; and so on.
The ancient world had an obsession with claiming that the entire world would constantly be destroyed and recreated – ie, recreated exactly the same, every time, with every critter exactly the same, and making exactly the same decisions. Reincarnation of individuals was the home game version of this “Great Cycle” idea. And again, it basically leads to the idea that you can do whatever you want, because ultimately it’s not your fault and nothing matters.
So yes, reincarnation is utterly the opposite of Christianity.
The world was created once by God. The world will end at a time of God’s choosing.
We all have one life each. What we do matters. It matters so much that God came into our world as true man and true God, to suffer and be resurrected once only, to redeem the sins of Adam, Eve, and the rest of us.
But our freewill matters so much that each of us can choose to refuse God’s gift of His life. Forever.
