Limit cardinal

From Academic Kids

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 λ is not a successor cardinal, i.e. we cannot "reach" λ by repeated successor operations. In precise terms λ is a limit cardinal if 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're not just defining something synonymous). In fact, all infinite cardinals are limit ordinals. However the concepts are closely tied via the aleph operation;α 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.

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 λ cannot be reached by repeated powerset operations, i.e. if for all κ < λ, 2κ < λ. Such a cardinal is also (with the axiom of choice) weak limit cardinal, as we'd expect from the names, since for any cardinal κ, κ+ ≤ 2κ (the proposition that the two are always equal 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's 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, ℵ0, is a limit cardinal of both "strengths." Actually, considerable contention, similar to the case of limit ordinals, exists as to whether or not zero and ℵ0 should be regarded as limits; some mathematicians insist that they be uncountable, others insist that they be infinite. In any case, the concept was invented to capture properties of ℵ0 relative to the finite cardinals.

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

<math>\beth_\omega = \bigcup_{n < \omega} \beth_n<math>

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 "trusty old union" 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 less than κ smaller cardinals. Such a cardinal is called a weakly (resp. strongly) inaccessible cardinal. The preceding examples both are singular cardinals of cofinality ω and hence they are not inaccessible.

0 is an inaccessible cardinal of both "strengths" (or at least, it captures the essential property of inaccessibility relative to the finite cardinals; some mathematicians will insist that such cardinals be uncountable); and as it turns out, standard Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) cannot even prove the existence of an inaccessible cardinal of either kind above ℵ0! These form the first in a hierarchy of large cardinals.

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