287: Unbounded Collection of Ordinal Numbers Is Not Set

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A description/proof of that unbounded collection of ordinal numbers is not set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of unbounded collection.
• The reader knows a definition of ordinal number.
• The reader admits the Burali-Forti theorem.

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Target Context

• The reader will have a description and a proof of the proposition that any unbounded collection of ordinal numbers is not any set.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

Any unbounded collection, $O$, of ordinal numbers, ordered by the $ϵ$ ordering, is not any set.

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2: Proof

$\cup O$ is the collection of all the ordinal numbers, because for any ordinal number, $o_1$, there is an ordinal number, $o_2\in O,o_1\in o_2$, so, $o_1\in \cup O$. By the Burali-Forti theorem, $\cup O$ is not any set. So, $O$ is not any set, because if $O$ was a set, $\cup O$ would a set by the union axiom.

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References

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