Question: When dealing with binary relations over infinite sets, it can be easy to accidentally conclude that a property holds of the relation that, while true for finite subsets of the infinite set, does not actually hold for the infinite set itself.
1. Let A be an infinite set and R be a binary relation over A. Suppose that for every finite subset A; ⊆ A, that R, restricted to A;, is a total order. Does this necessarily mean that A is a total order? If so, prove it. If not, find a counter example.
2. Let A be an infinite set and R be a binary relation over A. Suppose that for every finite subset A; ⊆ A, that R, restricted to A;, is a well order. Does this necessarily mean that A is a well order? If so, prove it. If not, find a counter example.