First order or second order? Tarsky or geometry?

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 the first-order logic, while philosophers and algebraists, and mathematicians in specific fields, may prefer the second-order logic and other more general logics (L_w1w, continuous logic, etc.). Macintyre's paper summarizes a frequent position of model-theorists, which is close to the position of physicists towards abstract mathematics: they feel they do not need rigorous abstract theory, but rather methods and techniques with practical use in their field.

Read the following papers:

G. S. Boolos: On second-order logic
J. Vaananen: Second-order logic and foundations of mathematics
A. Macintyre: Geometrical and set-theoretic aspects and prospects

Formulate your own position on this matter for a discussion.