Section 6: Finite Sets

We have not yet shown that the cardinality of a finite set is uniquely determined by the set... Even empirical demonstration would be difficult... One might, for instance, construct an experiment by taking a freight car full of marbles and hiring 10 different people to count them independently. If one thinks of the physical problems involved, it seems likely that the counters would not all arrive at the same answer. Of course, the conclusion one could draw is that at least one person made a mistake. But that mean assuming the correctness of the result one was trying to demonstrate empirically. An alternative explanation could be that there do exist bijective correspondences between the given set of marbles and two different sections of the positive integers. In real life, we accept the first explanation. We simply take it on faith that our experience in counting comparatively small sets of objects demonstrates a truth that holds for arbitrarily large sets as well. However, in mathematics... one does not have to take this statement on faith. If it is formulated in terms of the existence of bijective correspondences rather than in terms of the physical act of counting, it is capable of mathematical proof.

• There is no bijection of a set with its proper subset.
  • The cardinality of a finite set is uniquely determined.
  • $\mathbb{Z}_{+}$ is not finite.
• A subset of a finite set is finite, and if it is a proper subset its cardinality is less than that of the set.
• A criterion for a set to be finite.
  Let $A$ be nonempty. $A$ is finite iff there is a surjective function $f:S_{n}\rightarrow A$ for some $n$ iff there is an injective function $g:A\rightarrow S_{m}$ for some $m$ .