Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Successor_cardinal> ?p ?o. }
Showing items 1 to 24 of
24
with 100 items per page.
- Successor_cardinal abstract "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,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:and for λ an infinite limit ordinal,If β is a successor ordinal, then 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 is a limit cardinal.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:.".
- Successor_cardinal wikiPageID "375210".
- Successor_cardinal wikiPageRevisionID "546461628".
- Successor_cardinal hasPhotoCollection Successor_cardinal.
- Successor_cardinal subject Category:Cardinal_numbers.
- Successor_cardinal subject Category:Set_theory.
- Successor_cardinal type Abstraction100002137.
- Successor_cardinal type CardinalNumber113597585.
- Successor_cardinal type CardinalNumbers.
- Successor_cardinal type DefiniteQuantity113576101.
- Successor_cardinal type Measure100033615.
- Successor_cardinal type Number113582013.
- Successor_cardinal comment "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).".
- Successor_cardinal label "Następnik liczby kardynalnej".
- Successor_cardinal label "Successor cardinal".
- Successor_cardinal label "Sucessor cardinal".
- Successor_cardinal sameAs Następnik_liczby_kardynalnej.
- Successor_cardinal sameAs Sucessor_cardinal.
- Successor_cardinal sameAs m.020ycq.
- Successor_cardinal sameAs Q7632662.
- Successor_cardinal sameAs Q7632662.
- Successor_cardinal sameAs Successor_cardinal.
- Successor_cardinal wasDerivedFrom Successor_cardinal?oldid=546461628.
- Successor_cardinal isPrimaryTopicOf Successor_cardinal.