250: There Is No Set That Contains All Sets

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A description/proof of that there is no set that contains all sets

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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 set.

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Target Context

• The reader will have a description and a proof of the proposition that there is no set that that contains all the sets.

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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

There is no set that contains all the sets.

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2: Proof

Let us suppose that there was a set that contains all the sets, $C=\{S|S\text{ is any set}\}$. By the subset axiom, $S^{\prime }=\{s\in C|s\notin s\}$ would be a set. $S^{\prime }\in C$. If $S^{\prime }\in S^{\prime }$, $S^{\prime }\notin S^{\prime }$, and if $S^{\prime }\notin S^{\prime }$, $S^{\prime }\in S^{\prime }$, a contradiction.

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References

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