[FOM] Axiomatization of Reflective Sequences

Dmytro Taranovsky dmytro at mit.edu
Tue Feb 17 12:07:58 EST 2015

With regard to my previous FOM posting, Hugh Woodin has found an 
inconsistency in the key combinatorial principle (which I mentioned 
might be inconsistent).  His argument can be used to show that:

Theorem:  There is a definable predicate P such that for every set of 
ordinals S of order type at least (2^2^omega_1)^+ there are subsets T_1 
and T_2 of order type omega_2*2 such that such that P(T_1) <==> not P(T_2).

Here is one such P: P(S) iff {S_i:i<omega_2} agrees with 
{S_i:omega_2<=i<omega_2*2} with respect to all predicates that are 
definable in V_sup(S) using parameters that are subsets of omega_1.

The theorem can be proved by showing that if P(T) were true for all 
subsets of S of order type omega_2*2, then the elements of S are 
indiscernibles (for subsets of length omega_2), which can be used to get 
omega_1-Suslin determinacy (analogously to the proof of analytic 
determinacy from indiscernibles obtained from sharps), a contradiction.

The weakening (that I suggested as a starting point) of my attempted 
axiomatization of set theory with indiscernibles does not appear to be 
affected.  However, the theorem sharply limits the degree of infinitary 
indiscernibility possible, and without sufficient indiscernibility, an 
axiomatization would be too incomplete, and thus both less interesting 
and failing its purpose to make the notion appear concrete in V.

That said, there may be good axiomatizations that weaken invariance 
under subsequences or limit the invariance to a restricted class of 
predicates.  One can also axiomatize weaker notions, such as which 
properties a sequence of ordinals of length omega should satisfy if the 
sequence has sufficiently strong reflection properties.

A good axiomatization is important evidence that a particular extension 
to the language of set theory is unambiguous in V.  In fact, for a 
formalist, a good approximation to "well-defined" is: There are axioms 
for the notion that can be shown to be intuitively true and that resolve 
all major incompleteness associated with the notion.  Of course, under 
this criterion, even Sigma^2_1 truth has not yet been conclusively 
established as well-defined. (CH + Sigma^2_1-correctness of "ZFC + 
measurable Woodin cardinal" may correctly resolve the major Sigma^2_1 
incompleteness, but its theory has not yet been fully worked out and its 
truthfulness remains controversial.)

Dmytro Taranovsky

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