Re: Set Theory-Axiom of Congruence.


zuhair wrote:
> The Axiom of Congruence:
>
> The cardinality of any set other than the empty set and the set of all
> sets is always greater than the cardinality of any proper subset of it.

When you say greater, I assume you mean greater than or equal to. This
must be the cast, because two sets such as {1, 2, 3, 4,...} and {2, 3,
4, ...} can be put into bijective correspondence, but one is a proper
subset of the other.

I always thought that the statement "Set A has cadinality greater than
or equal to that of set B" was DEFINED to mean "There is an injective
map from B into A." If this is the definition you are using, then this
"axiom" can easily be proven. The inclusion map is an injection from a
subset of a set into that set.

.

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