On 4/11/2013 10:19 AM, Dan wrote: > I hold that ideally , when we don't know of the necessity of a (that > is , not (?a ) holds) , that we should know that it is necessarily so > that we don't know of the necessity of a , that is , (? not (?a )) . > This makes "classical logic + ?" non-monotonic .
The pragmatic philosophers, Pierce and James, had issues with representation of classical logic precisely because of some of the types of non-monotonicity in the wikipedia article.
> It also doesn't > prevent anyone from exploring various consequences of either CH or > (not CH) . My assumption was it is possible to formulate some > questions such that, due some inherent circularity , are rendered > unanswerable . Now, this leads to two possible situations : either > we know of a statement's truth or falsity, or we know that we can't > know of a statement's truth or falsity (as is the case with > superdeterminism) . It would be sadistic to assume any other > possibility (such as "we can't know , and we can't know that we can't > know). >
That makes sense.
I am afraid GCH may be precisely one of those statments rendered unanswerable by the inherent circularity issue.
Goedel's 1939 assumes denotations.
First-order logic assumes denotations.
So, the assumptions occur inside and outside of models.
Intuitionistic arithmetization methods are invoked to eliminate any defined constant symbols of the notion of set theory.
But, the entire sense of model theory is that truth derives from compositionality of terms and compositionality of formulas in relation to an assignment of extra-logical names. The manipulation of eliminating descriptive definitions defeats the model theory somewhat.
But, if one considers the extra-logical names, then one is faced with a consistent global labelling phenomenon in the sense of names as unique identifiers. Given names, permutations abound. But, what is assumed is a canonical sequential order corresponding to the assignment of names.
In a sense, then, no model of set theory can have greater plenitude than its ordinals. Relative to definability within the model, this yields V=OD (ordinal definability).
Kunen points out that V=OD requires one to know all the sets of V. So, it is second order.
Relative to predicative instantiation in second order, L=HOD (constructible universe = hereditarily ordinal definable).
GCH is true in L. But, if my assessment is reasonable, it is because of the issue of denotation. Note that this is also related to a property of a global canonical ordering. GCH has a close affinity with the axiom of choice.
So, these questions I have had with respect to denotation appear to be related to the classical logic itself. It may even be that "assuming choice" or "assuming choice via GCH" is equivalent to assuming CON(T) with respect to a theory.
The sense of this analysis is that GCH is a question so bound up with model theory that its independence corresponds with your statement of circularity.
I have no problem with circularity, however. It only needs to be introduced relative to an understanding of its epistemic role.