Re: SUO: Theory Query
Jon,
There are lots of things to worry about in mathematics, science, and
life. But there is a difference between worrying about the definition
of a mathematical construction and its theoretical implications and
worrying about how and whether it might be used or abused in practice.
> But when it comes to the quotient lattice,
> it is my experience that proving different
> axiom sets equivalent can be rather tricky,
> and so you may never know where your favorite
> axiom set is located in the lattice of theories.
> So I worry about that.
I agree that it is an issue that is worth being concerned about when
using the idea in practice. But that same question arises in slightly
different guises in every other approach -- formal or informal.
The reasons why I like the Lindenbaum lattice are that it brings these
questions into the open, it provides a systematic perspective on them,
it classifies the kinds of operations that may be performed on the
theories, and it unifies many diverse operations as different aspects
of a common underlying formalism.
Bottom line: The Lindenbaum lattice doesn't solve all the world's
problems, but it provides a systematic view of some important problems,
and it provides a convenient framework in which they can be addressed.
John Sowa