Section outline

  • 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 mathematics

    Formulate your own position on this matter for a discussion.