Sunday, March 22, 2009

After having thought about it, I am becoming more convinced that the correct approach to the issues mentioned in these two posts is to accept the line of reasoning that deduces an enumerator's truth from the truth of all its consequences, thus (as mentioned) inviting paradox.

Consider the original stated purpose of my investigation: to examine which statements are probabilistically justifiable. To probabilistically accept one of the enumerators simply because (so far) all of its implications appear to be true would be a violation of the semantics, unless the truth value of an enumerator really was just the conjunction of the enumerated statements. So to satisfy my original purpose, such a thing is needed.

As I said, paradox can be avoided by allowing enumerators to be neither true nor false. In particular, it is natural to suppose a construction similar to Kripke's least-fixed-point theory of truth: an enumerator is "ungrounded" if it implies some chain of enumerators which never bottoms out to actual statements; ungrounded enumerators are neither true nor false.

The problem is that we will now want to refer to the ungrounded-ness of those sentences, since it appears to be a rather important property. For this we need to augment the language. This can be done in multiple ways, but it will ultimately lead to a new reference failure. And filling in that hole will lead to yet another hole to fill. In general I would deal with this by using my theory that almost works. This entails building an infinite hierarchy of truth values which starts:

True, False, Meaningless1, Meaningless2, Meaningless3, MeaninglessInfinity, MeaninglessInfinity+1...

I am generally interested in investigating whether this hierarchy is equal in power to the usual Tarski hierarchy, namely,

True1, True2, True3, ... TrueInfinity, TrueInfinity+1, ...

The difference basically springs out of the use of a loopy truth predicate (a truth predicate that can apply truly to sentences which contain the same truth predicate). Asking for a single truth predicate appears to force the existence of infinitely many meaninglessness predicates. Is a loopy truth predicate automatically more powerful? Not obviously so: the loopy truth predicate will have a maximal amount of mathematical structure that it implies (the least fixed point), which does not even include uncountable entities. The Tarski hierarchy will continue into the uncountables, and further. (I should note that that statement is not universally accepted... but if someone suggests that the Tarski hierarchy should stop at some well-defined level, can't I just say, OK, let's make a truth predicate for that level? Shy not keep going? And we don't actually need an uncountable number of truth predicates: we need an ordinal notation that can refer to uncountable ordinals, and we just make the truth predicate take an ordinal as an agrument.)

So the question is: is there an isomorphism between the mathematical structures captured by the Tarski heirarchy of truth, and my hierarchy of nonsense.

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