Does there exist a sequence of sets such that the sequences of their cardinalities is strictly decreasing?
More explicitly, does there exist a sequence of sets $S_1,S_2…$ such that for each $i$ there exists an injection from $S_{i+1}$ into $S_i$ yet $S_i$ and $S_{i+1}$ is no bijection?
In other words, each $S_{i+1}$ is strictly smaller (in cardinality) than $S_i$.
Best Answer
No: cardinalities are well-ordered, and no well-order admits an infinite descending chain. Such a chain would be a non-empty set with no least element.