Re: Goedel incompl. theorems


waveletter skrev:


Because of (a) and (d), some mathematicians and philosophers (e.g.
Quine) assert that 2nd-order logic is improper for mathematics and
science. It's proof theory is too problematic. But, what ordinary
mathematicians use is essentially 2nd-order logic without the
formalism. The power and commonality of the 2nd-order theory, despite
its ugly proof theory, are still attractive. Some people try to find
limited versions of 2nd-order logic that have a decent proof theory but
retain enough power to derive all of conventional mathematical
analysis. Boolos himself was one of these. If you're interested in this
pursuit, you might seek out:


What do you mean with "But, what ordinary mathematicians use is
essentially 2nd-order logic without the formalism"? Is not most of
ordinary mathematics possible to formalize in ZFC, which is a first
order theory?

A last question, is not Second Order Arithmetic most often formalized
using first order logic?


3. Simpson, "Subsystems of Second Order Arithmetic," Berlin: Springer,
1999.

4. Finally, check out the Wikipedia page:
http://en.wikipedia.org/wiki/Second-order_arithmetic

Hope this helps!
--Ron Allen

.

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