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## Main Question or Discussion Point

Does most of mathematics use 2nd order logic? If so would studying the foundations of mathematics involve mostly using 2nd order logic?

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Does most of mathematics use 2nd order logic? If so would studying the foundations of mathematics involve mostly using 2nd order logic?

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CRGreathouse

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There's value in simplicity, so while many theories can be expressed in second-order logic I generally see them as being embedded in first-order logic. But there's no escaping real complexity; ZF in first-order logic has infinitely many axioms, so perhaps one would argue that its second-order formulation is 'no worse'.

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