Cardinality - Infinite Sets

Infinite Sets

Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel.

The reason for this is that the various characterizations of what it means for set A to be larger than set B, or to be the same size as set B, which are all equivalent for finite sets, are no longer equivalent for infinite sets. Different characterizations can yield different results. For example, in the popular characterization of size chosen by Cantor, sometimes an infinite set A is larger (in that sense) than an infinite set B; while other characterizations may yield that an infinite set A is always the same size as an infinite set B.

For finite sets, counting is just forming a bijection (i.e., a one-to-one correspondence) between the set being counted and an initial segment of the positive integers. Thus there is no notion equivalent to counting for infinite sets. While counting gives a unique result when applied to a finite set, an infinite set may be placed into a one-to-one correspondence with many different ordinal numbers depending on how one chooses to "count" (order) it.

Additionally, different characterizations of size, when extended to infinite sets, will break different "rules" which held for finite sets. Which rules are broken varies from characterization to characterization. For example, Cantor's characterization, while preserving the rule that sometimes one set is larger than another, breaks the rule that deleting an element makes the set smaller. Another characterization may preserve the rule that deleting an element makes the set smaller, but break another rule. Furthermore, some characterization may not "directly" break a rule, but it may not "directly" uphold it either, in the sense that whichever is the case depends upon a controversial axiom such as the axiom of choice or the continuum hypothesis. Thus there are three possibilities. Each characterization will break some rules, uphold some others, and may be indecisive about some others.

If one extends to multisets, further rules are broken (assuming Cantor's approach), which hold for finite multisets. If we have two multisets A and B, A not being larger than B and B not being larger than A does not necessarily imply A has the same size as B. This rule holds for multisets that are finite. Needless to say, the law of trichotomy is explicitly broken in this case, as opposed to the situation with sets, where it is equivalent to the axiom of choice.

Dedekind simply defined an infinite set as one having the same size (in Cantor's sense) as at least one of its proper parts; this notion of infinity is called Dedekind infinite. This definition only works in the presence of some form of the axiom of choice, however, so will not be considered to work by some mathematicians.

Cantor introduced the above-mentioned cardinal numbers, and showed that (in Cantor's sense) some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers (ℵ₀).