Let's compare the cardinalities of the given sets:
1. The set of real numbers R has the cardinality of the continuum, which is uncountably infinite and has the same cardinality as the points on a line.
2. The set of all functions from R to {0,1} has a cardinality strictly greater than the cardinality of the continuum. This is because there are 2^c (where c is the cardinality of R) different functions from R to {0,1}. In other words, it has a larger cardinality than the set of real numbers.
3. The set of all finite subsets of natural numbers has a countably infinite cardinality, which is the same as the cardinality of the natural numbers themselves.
4. The set of all finite-length binary strings also has a countably infinite cardinality. This is because for each positive integer n, there are 2^n different binary strings of length n.
Comparing these, the set of all functions from R to {0,1} has the greatest cardinality among the given sets.
