Following ordinal

  • Thread starter ibc
  • Start date


(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"
Last edited:
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".

Want to reply to this thread?

"Following ordinal" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving