FOM: RE: Re: SOL confusion

[Matt Insall]

Harvey:
So in a sound and complete finitary deductive calculus, semantic
compactness is equivalent to strong completeness.
<SNIP>
Yes, but the hypothesis of completeness is far too strong in the context of
SOL. The set of SOL validities is nowhere near even recursively enumerable.
<SNIP>
SOL is well known not to be semantically compact, and also any complete
deductive system for SOL must be grossly unreasonable.
Matt:
Right. Now, take any complete deductive system for SOL (which is therefore
``grossly unreasonable''). [If someone wants to claim they have such a
deductive system, I am willing to let them claim it.] If it is sound, then
it fails strong completeness. Thus, second order logic not only is
incomplete, but it is grossly incomplete, for even allowing someone a
complete deductive calculus for SOL with standard semantics, there are
results they cannot prove. I knew that I had misgivings about how much of
its own ``truth'' SOL can discover, and I knew there must be a careful
formulation of what was bothering me that I was not expressing clearly.
Thank you for your patience on this particular point.
Harvey:
What is "it"? The correct theorem above requires no form of AxC. However,
compactness for arbitrary sets of sentences of FOL does require some form
of AxC.
Matt:
You are of course, correct.
Harvey:
We have been talking about versions deductive SOL endlessley on the FOM,
none of which are "worthy".
Matt:
Ah yes. I see now that you do not allow the validities as axioms because
they form a non-r.e. set. The validities are only the consequences of the
empty set of ``logical axioms'', however. For ``worthiness'', I wish to
allow a non-r.e. set of ``logical axioms''. (I think it is immaterial
whether anyone will ever obtain such a set of ``logical'' axioms, for my
purpose.) Then for any claim that ``more of the truth'' can be obtained
using SOL (especially deductive SOL), we have a very generous hypothesis and
a very tight conclusion. Does this now seem more along the lines of what
you mean when you say that SOL is not a deductive system, it is a semantic
system. (I do not recall when I saw this in one of your posts, so although
I attribute it to you, I am not certain I should put quotes around it, in
case I am in error.) I agree that it is fundamentally a semantic system,
because no matter how much leeway is given to a finitistic and mechanical
interpretation of provability (even so far as to allow it access to all the
validities as ``oracles'' - is that not what they are then called?), there
are always unprovable results.
 >One might object, saying that it is unclear how to determine when a
 >deductive system is ``worthy''. But in fact, is that not the common design
 >when someone seriously considers devising a deductive calculus?
Harvey:
No.
Matt:
I guess that in computer science, one may devise a deductive calculus merely
to obtain a new language, or to demonstrate another limitation on some
particular conceptual language design. However, what, in pure mathematics,
is valuable about a deductive system that is not complete, as opposed to one
that is complete? Is Montague's book, which you mentioned in an earlier
post, a good source for information about the purpose of devising incomplete
deductive calculi? Also, I do not see how an incomplete deductive system is
any more useful in the philosophy of mathematics or the philosophy of
science than a complete calculus.
Dr. Matt Insall
http://www.umr.edu/~insall