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Encyclopedia > Successor cardinal

In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality (a bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above; in the style of Hilbert's Hotel Infinity). Using the von Neumann cardinal assignment and the axiom of choice (AC), this successor operation is easy to define: for a cardinal number κ we have Alternative meaning: number of pitch classes in a set. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... A bijective function. ... Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 – 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ... The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

kappa^+ = |inf { lambda in ON  | kappa < |lambda| }| ,

where ON is the class of ordinals. That is, the successor cardinal is the cardinality of the least ordinal into which a set of the given cardinality can be mapped one-to-one, but which cannot be mapped one-to-one back into that set.


That the set above is nonempty follows from Hartogs' theorem, which says that for any well-orderable cardinal, a larger such cardinal is constructible. The minimum actually exists because the ordinals are well-ordered. It is therefore immediate that there is no cardinal number in between κ and κ+. A successor cardinal is a cardinal which is κ+ for some cardinal κ. In the infinite case, the successor operation skips over many ordinal numbers; in fact, every infinite cardinal is a limit ordinal. Therefore, the successor operation on cardinals gains a lot of power in the infinite case (relative the ordinal successorship operation), and consequently the cardinal numbers are a very "sparse" subclass of the ordinals. We define the sequence of alephs (via the axiom of replacement) via this operation, through all the ordinal numbers as follows: In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... A limit ordinal is an ordinal number which is not a successor ordinal. ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ...

aleph_0 = omega
aleph_{alpha+1} = aleph_{alpha}^+

and for λ an infinite limit ordinal,

aleph_{lambda} = bigcup_{beta < lambda} aleph_beta

If β is a successor ordinal, then aleph_{beta} is a successor cardinal. Cardinals which are not successor cardinals are called limit cardinals; and by the above definition, if λ is a limit ordinal, then aleph_{lambda} is a limit cardinal. When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one. ... In mathematics, limit cardinals are a type of cardinal number. ...


The standard definition above is restricted to the case when the cardinal can be well-ordered, i.e. is finite or an aleph. Without the axiom of choice, there are cardinals which cannot be well-ordered. Some mathematicians have defined the successor of such a cardinal as the cardinality of the least ordinal which cannot be mapped one-to-one into a set of the given cardinality. That is:

kappa^+ = |inf { lambda in ON  | |lambda| nleq kappa }| .

See also

In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets which are equinumerous). ...

References

  • Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
  • Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

  Results from FactBites:
 
Infinite Ink: Cardinal Numbers (1276 words)
Since all well-ordered cardinals are ordinals, sometimes I use ordinal notation for a cardinal or vice versa.
A cardinal that is not a finite cardinal is an infinite cardinal.
In ZF it is possible to have a Dedekind-finite cardinal that is not a finite cardinal.
Limit cardinal - Wikipedia, the free encyclopedia (549 words)
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.
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).
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.
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