Re: SUO: Theory Query
Chris,
A few remarks:
1. I really don't care about the continuum hypothesis, since I have
long had doubts about "Cantor's paradise" as Hilbert called it.
2. I agree that one can define a one-to-one mapping between the
two lattices.
3. But some mappings are more "natural" than others, in the sense
that you can also apply them to finite subsets (where simple
counting is a good measure). An example is the set of primes,
which become consistenly more sparse as you go to larger integers.
4. The work on measure theory in analysis is an example of the way
one can define "natural" measures to distinguish the size of sets
whose cardinality in Cantor's sense happens to be identical.
5. I'll admit that I don't have a good definition of "measure" for
the two lattices we have been talking about, and I don't care to
take the time to bother defining some kind of measure. But I have
a feeling that some "natural" measure would make the lattice of
all subsets significantly "bigger" than the lattice of deductive
closures.
6. I agree with you that when we consider the complete deductive
closures, the relation of "A implies B" is equivalent to
"B is a subset of A". But for most applications, we will only
be looking at finite axiomatizations for which implication is
the primary relation we have to work with.
7. And I agree that most current textbooks restrict the term
"tautology" in the way that you prefer. But as I have said
many times, most current textbooks are woefully deficient in
the way they teach logic -- and the current distaste for logic
among significant numbers of programmers is just one symptom
of that deficiency. (Of course, the current use of "tautology"
is a rather trivial issue, but it arises from an outmoded way
of teaching proof theory.)
John