FOM: RE: Re: SOL confusion

Matt Insall montez at
Mon Sep 11 15:37:28 EDT 2000

So in a sound and complete finitary deductive calculus, semantic
compactness is equivalent to strong completeness.
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.
SOL is well known not to be semantically compact, and also any complete
deductive system for SOL must be grossly unreasonable.
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.
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.
You are of course, correct.
We have been talking about versions deductive SOL endlessley on the FOM,
none of which are "worthy".
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?
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

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