First order or second order?
Section outline
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Natural axiomatizion of both set theory and arithmetics is "second-order' in the sense that both the REPLACEMENT in set theory and INDUCTION in arithmetics quantifies over collections of objects of the universe.
However, second-order axiomatization has problems related to completentess, and therefore does not deliver "what it promises".
Usually, mathematicians favour first-order logic, while philosophers may argue for second-order.
Read the two following papers:
G. S. Boolos: On second-order logic
J. Vaananen: Second-order logic and foundations of mathematicsFormulate your own position on this matter for a discussion.