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.

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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; <math>\aleph_{\alpha}</math> 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 < λ. Such a cardinal is also a weak limit cardinal, as we would expect from the names, since for any cardinal κ, κ ≤ 2. (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, <math>\aleph_0</math>, is a limit cardinal of both "strengths". An obvious way to construct more limit cardinals of both strengths is via the union operation: <math>\aleph_{\omega}</math> is a limit cardinal, defined as the union of all the alephs before it; and in general <math>\aleph_{\lambda}</math> 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>
  • Izolovaný kardinál a limitní kardinál jsou pojmy z teorie množin, které rozdělují kardinální čísla na dvě disjunktní třídy podle postavení v hierarchii kardinálů.
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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.
  • Izolovaný kardinál a limitní kardinál jsou pojmy z teorie množin, které rozdělují kardinální čísla na dvě disjunktní třídy podle postavení v hierarchii kardinálů.
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  • Limit cardinal
  • Izolovaný a limitní kardinál
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