Limit cardinal

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In mathematics, limit cardinals are a type of cardinal number.

With the cardinal successor operation defined, we can define a limit cardinal in analogy to that for limit ordinals: λ is a (weak) limit cardinal if and only if λ is neither a successor cardinal nor zero, i.e. we cannot "reach" λ by repeated successor operations. In precise terms λ is a limit cardinal if and only if there is a κ < λ and for all κ < λ, κ⁺ < λ. Despite the similarity in terminology and concept with limit ordinal, being a limit cardinal is a much stronger condition, because the cardinal successor operation is much more powerful, in the infinite case, than the ordinal successor operation (so we are not just defining something synonymous). In fact, any initial ordinal of an infinite cardinal is a limit ordinal; and if the axiom of choice holds, every infinite cardinal has such an initial ordinal. However the concepts are closely tied via the aleph operation; $\aleph_{\alpha}$ is a successor cardinal if and only if α is a successor ordinal, hence also a limit cardinal if and only if α is a limit ordinal or zero.

The axioms of set theory give us another operation, the power set operation, that always gives a set of strictly larger cardinality; this motivates the following definition: a cardinal λ is a strong limit cardinal if and only if λ cannot be reached by repeated powerset operations, i.e. if and only if there is a κ < λ and for all κ < λ, 2^κ < λ. Such a cardinal is also a weak limit cardinal, as we would expect from the names, since for any cardinal κ, κ⁺ ≤ 2^κ. (The proposition that this last "≤" is really "=" in the infinite case is precisely the generalized continuum hypothesis. Perhaps central to the debate is how much "extra power" the successor operation acquires in the infinite case; it is obvious that in the finite case, powerset skips over many more cardinal numbers than successorship does, yet the infinite case renders many "big" operations such as multiplication as trivial "maximum" operations, while exponentiation still manages to increase cardinality. It is interesting to see where successorship lies in this "spectrum of operations.").

The first infinite cardinal, $\aleph_0$ , is a limit cardinal of both "strengths".

An obvious way to construct more limit ordinals of both strengths is via the union operation: $\aleph_{\omega}$ is a limit cardinal, defined as the union of all the alephs before it; and in general $\aleph_{\lambda}$ for any limit ordinal λ is a limit cardinal. Similarly, we do the same with beth numbers ( $\beth$ is beth, the second letter of the Hebrew alphabet) to get strong limit cardinals such as

$\beth_\omega = \bigcup_{n < \omega} \beth_n$

[edit] The notion of inaccessibility and large cardinals

The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above. But the "union operation" always manages to help us out and gives us another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). So this is not the last word on inaccessibility: mathematicians, of course, always like to "jump up" a level. We can make life difficult even with the union operation by using cofinality. For a weak (resp. strong) limit cardinal κ we can demand that cf(κ) = κ (i.e. κ be regular) so that κ cannot be expressed as a sum (union) of fewer than κ smaller cardinals. Such a cardinal is called an weakly (resp. strongly) inaccessible cardinal. The preceding examples both are singular cardinals of cofinality ω and hence they are not inaccessible.

$\aleph_0$ would be an inaccessible cardinal of both "strengths" except that the definition of inaccessible requires that they be uncountable. Standard Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) cannot even prove the existence of an inaccessible cardinal of either kind above $\aleph_0$ ! These form the first in a hierarchy of large cardinals.