251: Collection of Sets That Are of Non-0 Cardinality Is Not Set

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A description/proof of that collection of sets that are of non-zero cardinality 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 set.
• The reader knows a definition of cardinal number.
• The reader admits the proposition that there is no set that that contains all the sets.

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Target Context

• The reader will have a description and a proof of the proposition that for any non-0 cardinal number, the collection of the sets that are of the cardinality 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

For any non-0 cardinal number, $c$, the collection of sets, $C=\{S|cardS=c\}$ where $cardS$ is the cardinal number of $S$ is not any set.

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2: Proof

Let us suppose that $C$ was a set. For any set, $S^{\prime }$, $S^{\prime }\in S$ for an $S\in C$, because for any $S\in C$, if $S^{\prime }\notin S$, there would be a set, $S^{″}\in S$ because $0<cardS$, and the new set, $S^{‴}=(S\setminus \{S^{″}\})\cup \{S^{\prime }\}$, would be a set of the cardinality, $c$, by the pairing axiom, the subset axiom, and the union axiom. By the union axiom, $\cup C$, which means ${\cup }_{S\in C}S$, would be a set, but that would contain all the sets, which would not be any set, by the proposition that there is no set that that contains all the sets, a contradiction.

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References

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