324: Descending Sequence of Ordinal Numbers Is Finite

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A description/proof of that descending sequence of ordinal numbers is finite

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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 ordinal number.
• The reader admits the proposition that any collection of ordinal numbers has the smallest element.

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Target Context

• The reader will have a description and a proof of the proposition that any descending sequence of ordinal numbers is finite.

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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 descending sequence, $o_0,o_1,...$ where $...\in o_2\in o_1\in o_0$, of ordinal numbers finishes at an $o_n$.

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2: Proof

By the proposition that any collection of ordinal numbers has the smallest element, there is the smallest element of $\{o_i\}$, which is $o_n$. Then, there is no $o_{n+1}$, because it would be a smaller element, a contradiction.

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References

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