FOM: 2nd order logic

[Karlis Podnieks]

My restricted (old Hilbert-style formalist) mind does not allow
to adopt any "second order" notions without a definite subsystem
of (first order!) set theory behind them. Of course, the
well-known second order arithmetic satisfies this condition. But
the so-called general "second order logics" try to discuss
"objects" and "properties of objects" without introducing a
full-fledged notion of "sets of objects", "sets  a s  objects",
"sets of sets of objects" etc. How can a mathematician stop at
this point? For me, the best 2nd order logic is ZF (or
Ackermann's set theory, which is equivalent to ZF, but has more
elegant axioms, see my web-site).

Karlis Podnieks
http://www.ltn.lv/~podnieks/
University of Latvia, Institute of Mathematics and Computer
Science