Does Every Ordinal Have a Following Cardinal?

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SUMMARY

Every cardinal has a following cardinal, defined as the minimal cardinal that is strictly larger. This conclusion is based on the definition of cardinals as a special type of ordinals. The process works because any initial segment of the class of ordinals is a set, and ordinals are well-ordered. If the axiom of choice is not assumed, the term "well-ordered cardinal" must be used instead of "cardinal".

PREREQUISITES
  • Understanding of cardinal and ordinal numbers in set theory
  • Familiarity with the axiom of choice in mathematics
  • Knowledge of well-ordered sets and their properties
  • Basic concepts of mathematical logic and proofs
NEXT STEPS
  • Study the implications of the axiom of choice on cardinality
  • Explore the properties of well-ordered cardinals
  • Research the relationship between cardinals and ordinals in set theory
  • Examine the concept of initial segments in well-ordered sets
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Mathematicians, logicians, and students studying set theory, particularly those interested in the properties of cardinals and ordinals.

ibc
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(not assuming any kind of continuum hypothesis of course) does every cardinal have a following one, i.e a minimal cardinal that is strictly larger?edit: ops, the title should be "following cardinal"
 
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Yes (assuming the definition of cardinals as a special type of ordinals), it's the minimum of the class of ordinals with greater cardinality than your cardinal (this works because any initial segment of the class of ordinals is a set, and because the ordinals are well-ordered). If you don't want to assume the axiom of choice, you'll have to replace "cardinal" with "well-ordered cardinal".
 

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