irreconcilable positions etc are so strongly opposed to each other that it is not possible for them to reach an agreement
The differences between the landowners and the conservationists were irreconcilable from the start.
if two beliefs or ideas are irreconcilable, it is not possible to believe both of them
This belief was irreconcilable with the Church's doctrine of salvation.
strong disagreements between two people who are married, given as a legal reason for getting a divorce