How is uncountability characterized in second order logic?

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SUMMARY

Uncountability in second order logic is characterized by the inability to establish a one-to-one correspondence between a set and its proper subsets. A set is defined as infinite if it can be matched with one of its proper subsets. Countably infinite sets can be paired with all their infinite subsets, while uncountably infinite sets cannot be matched in this manner, establishing a clear distinction between the two types of infinity.

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How is uncountability characterized in second order logic?
 
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A set is infinite if it can be put in one-to-one correspondence with one of its proper subsets. A set is countably infinite if it is infinite and it can be put in one-to-one correspondence with every one of its infinite subsets. A set is uncountably infinite if it is infinite but not countably infinite.
 

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