# Does Every Ordinal Have a Following Cardinal?

• ibc
In summary, an ordinal number represents the position or order of an object, while a cardinal number represents the quantity or number of objects in a set. Not every ordinal has a following cardinal and they cannot be used interchangeably. However, ordinal numbers can be converted to cardinal numbers by counting the objects in a set. There are exceptions to the rule that every ordinal has a following cardinal, such as "half" and some infinite ordinals like "infinity".
ibc
(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".

## 1. What is an ordinal and a cardinal number?

An ordinal number is a number that represents the position or order of an object in a sequence. For example, first, second, third, etc. A cardinal number, on the other hand, represents the quantity or number of objects in a set. For example, one, two, three, etc.

## 2. Does every ordinal have a following cardinal?

No, not every ordinal has a following cardinal. Ordinal numbers can continue infinitely, while cardinal numbers only go up to a certain point. For example, there is no cardinal number that follows the ordinal number "infinity".

## 3. What is the relationship between ordinal and cardinal numbers?

Ordinal and cardinal numbers are related in that ordinal numbers can be converted to cardinal numbers by counting the number of objects in a set. For example, the ordinal number "third" can be converted to the cardinal number "3" if there are 3 objects in the set.

## 4. Can ordinal and cardinal numbers be used interchangeably?

No, ordinal and cardinal numbers have different purposes and cannot be used interchangeably. Ordinal numbers are used to represent order or position, while cardinal numbers represent quantity or number.

## 5. Are there any exceptions to the rule that every ordinal has a following cardinal?

Yes, there are some exceptions to this rule. For example, the ordinal number "half" does not have a corresponding cardinal number. Also, some infinite ordinals do not have a corresponding cardinal number, such as "infinity" or "aleph-null".

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