How is uncountability characterized in second order logic?

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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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