Looks like I have three roles here.
1. Very lately, some real new content that actually investigates some generally understandable aspects of “intrinsic maximality”. This has led rather nicely to legitimate foundational programs of a generally understandable nature, involving new kinds of investigations into decision procedures in set theory.
2. Attempts to direct the discussion into more productive topics. Recall the persistent subject line of this thread! The last time I tried this, I got a detailed response from Peter which I intended to answer, but put 1 above at a higher priority.
3. And finally, some generally understandable commentary on what is both not generally understandable and having no tangible outcome.
This is a brief dose of 3.
QUOTE FROM BSL PAPER BY MR. ENERGY (jointly authored):
The approach that we present here shares many features, though not all, of Goedel’s program for new axioms. Let us brieﬂy illustrate it. The Hyperuni- verse Program is an attempt to clarify which ﬁrst-order set-theoretic state- ments (beyond ZFC and its implications) are to be regarded as true in V , by creating a context in which diﬀerent pictures of the set-theoretic universe can be compared. This context is the hyperuniverse, deﬁned as the collection of all countable transitive models of ZFC.
DIGRESSION: The above seems to accept ZFC as “true in V”, but later discussions raise issues with this, especially with AxC.
So here we have the idiosyncractic propogandistic slogan “HP” for
And we have the DEFINITION of the hyperuniverse as
**the collection of all countable transitive models of ZFC**
QUOTE FROM THIS MORNING BY MR. ENERGY:
That is why it is quite inappropriate, as you have done on numerous occasions, to refer to the HP as the study of ctm’s, as there is no need to consider ctm’s at all, and even if one does (by applying LS), the properties of ctm’s that results are very special indeed, far more special than what a full-blown theory of ctm’s would entail.
If it is supposed to be “inappropriate to refer to the HP as the study of ctm’s”, and “no need to consider ctm’s at all”, then why coin the term Hyperuniverse Program and then DEFINE the Hyperuniverse as the collection of all countable transitive models of ZFC???
THE SOLUTION (as I suggested many times)
Stop using HP and instead use CTMP = countable transitive model program. Only AFTER something foundationally convincing arises, AFTER working through all kinds of pitfalls carefully and objectively, consider trying to put forth and defend a foundational program.
In the meantime, go for a “full-blown theory of ctm’s” (language from Mr. Energy) so that you at least have something tangible to show for the effort if and when people reject your foundational program(s).
GENERALLY UNDERSTANDABLE AND VERY DIRECT PITFALLS IN USING INTRINSIC MAXIMALITY
It is “obvious” from intrinsic maximality that the GCH fails at all infinite cardinals because of “width considerations”.
This “refutes” the continuum hypothesis. This also “refutes” the existence of -extendible cardinals, since they imply that the GCH holds at some infinite cardinals (Solovay).
LESSONS TO BE LEARNED
You have to creatively analyze what is wrong with the above use of “intrinsic maximality”, and how it is fundamentally to be distinguished from other uses of “intrinsic maximality” that one is putting forward as legitimate. If this can be done in a suitably creative and convincing way, THEN you have at least the beginnings of a legitimate foundational program. WARNING: if the distinction is drawn too artificially, then you are not creating a legitimate foundational program.