Cardinality of the set of all ordinals.

In summary, the cardinality of the set of all ordinals is an infinite number, denoted by the symbol "aleph-null" or ℵ<sub>0</sub>, which represents the first infinite cardinal number in the series of infinite cardinal numbers. It is greater than the cardinality of the set of natural numbers, but smaller than the cardinality of the set of real numbers. It cannot be calculated and has important implications in set theory and the study of infinite sets.
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TylerH
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What would the cardinality of the set of all ordinal numbers be? Is it even known or does the question even make sense in the case of such a weird, almost paradoxical set?
 
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The class of all ordinals doesn't form a set. There are simply too many ordinals to enclose them all in a set. Rather, the class of all ordinals forms a proper class. In ZFC set theory, there is no way to assign a cardinality to a proper class.

This should be interesting for you: http://en.wikipedia.org/wiki/Burali-Forti_paradox
 

What is the definition of the cardinality of the set of all ordinals?

The cardinality of the set of all ordinals is the number of elements in the set. It represents the size or magnitude of the set, regardless of the actual values of the elements.

Is the cardinality of the set of all ordinals a specific number?

No, the cardinality of the set of all ordinals is not a specific number. It is an infinite number, denoted by the symbol "aleph-null" or ℵ0, which represents the first infinite cardinal number in the series of infinite cardinal numbers.

How does the cardinality of the set of all ordinals compare to other infinite sets?

The cardinality of the set of all ordinals is greater than the cardinality of the set of natural numbers (ℵ0), but it is smaller than the cardinality of the set of real numbers (ℵ1). This means that there are more ordinals than natural numbers, but there are still more real numbers than ordinals.

Can the cardinality of the set of all ordinals be calculated?

No, the cardinality of the set of all ordinals cannot be calculated. It is an uncountable infinity, meaning that its elements cannot be put into a one-to-one correspondence with the natural numbers. Therefore, it is impossible to determine its exact value.

What is the significance of the cardinality of the set of all ordinals in mathematics?

The cardinality of the set of all ordinals has important implications in set theory and the study of infinite sets. It helps to define and compare different levels of infinity and is a fundamental concept in understanding the properties and behaviors of infinite sets.

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