FOM: generic absoluteness and CH
steel@math.berkeley.edu
steel at math.berkeley.edu
Fri Dec 19 13:13:31 EST 1997
I would like to present some technical problems in set theory, together
with arguments that solving them might well lead to a rational decision on
CH.
In outline, the argument which motivates these problems is the
following: although the theory we get from large cardinal axioms does not
seem to decide CH ( "large cardinal axiom" is inherently vague, so this is
not a theorem ), it does seem to decide all natural statements of 2nd
order arithmetic, and much more. Indeed CH, which is Sigma^2_1, is just on
the other side of the boundary, in terms of logical complexity, of the
class of statements for which large cardinal axioms seem to yield a
complete theory. Generic absoluteness theorems are key indicators of the
completeness of the theory in the language of 2nd order arithmetic and
beyond that we get from large cardinal axioms. These say, in effect, that
there are no independence results in this area provable by forcing. The
problems I wish to suggest involve extensions of these generic
absoluteness theorems to the realm of arbitrary Sigma^2_n statements. Such
extensions would indicate a "complete picture" at the Sigma^2_n level
analogous to the complete picture at the Sigma^1_n level we get from large
cardinal axioms.
I will break the detailed discussion into separate posts:
I.The consistency strength hierarchy
II. Large cardinal hypotheses as a preferred way of climbing the
consistency strength hierarchy
III.The completeness of large cardinal hypotheses in the realm of
statements about arbitrary reals; generic absoluteness theorems and
conjectures.
I. The consistency strength hierarchy.
Let T be a consistent theory in the language of set theory having a
modicum of strength; then every Pi^0_1 consequence of T is true. (This,
of course, helps motivate Hilbert's program.) If T is in addition
axiomatizable, then Con(T) is a true Pi^0_1 sentence not among the
consequences of T. ( This, of course, did great damage to Hilbert's
program.) Let P_T be the set of all Pi^0_1 consequences of T. In place of
the Hilbert picture, in which all P_T are the set of Pi^0_1 truths, we
have the Godel picture, in which there is an unending hierarchy of P_T's.
It is a truly remarkable phenomenon, which Godel may not have known, that
NATURAL consistent, axiomatizable extensions of ZFC appear to be
prewellordered by the relation
T \le S iff P_T is a subset of P_S.
( A prewellorder is a wellfounded, linear quasiorder. The mere fact of
linearity is remarkable.) One must emphasize "natural" here, because one
can construct a pair of self-referential sentences which yield
incomparable T's in this ordering. ( Like the proof of the double
recursion theorem. I learned this from Harvey, who said he could embed all
manner of complicated orderings. Does anyone have a reference?)
Nevertheless, so far as we (I?) know all theories which have something
remotely like an "idea" to them seem to be comparable with one another in
this ordering. This has actually been proved in many cases, for theories
whose ideas and motivations have nothing at all to do with one another.
This prewellordering by consistency strength extends to theories much
weaker than ZFC.
I should remark that for natural T and S, P_T subset P_S iff Peano
Arith. proves Con(S) implies Con(T). ( So far as I know, anyway.)
This prewellordering of natural consistency strengths seems to me very
strong evidence of something objective about the process of extending ZFC
to stronger theories, at least as it has been practiced so far. If we were
just randomly making things up, wouldn't we tend to go off in incomparable
directions? At the level of Pi^0_1 sentences we have not. I wonder what
others on the list make of this phenomenon. Is it just a manifestation of
lack of imagination or inbreeding? (Is it real- what examples of
incomparable consistency strengths do you know?)
Let me emphasize the scope of this phenomenon with an example.
If T asserts that GCH fails at some singular strong limit cardinal, S
asserts that there is a saturated ideal on omega_1, and U asserts that the
combinatorial principle square kappa-finite fails at some singular kappa
such that alpha<kappa implies alpha^omega < kappa, then: P_T subset P_S
subset P_U. (The inclusions are proper.) On their surfaces, T,S, and U
have little to do with one another. Why are their consistency strengths
linearly ordered? Such examples can be multiplied.
A caution: there are many natural pairs S,T which are not known at the
moment to be comparable in consistency strength. The only way we have to
prove S comparable with T is to actually determine their ordering, and
this has not yet been done in many cases.
If the prewellordering of "natural" consistency strengths is real, then
one would like to prove this somehow. Here is a partial result which seems
relevant. In practice, one calibrates the consistency strength of T by
identifying the weakest large cardinal axiom A such that any model of A
has a generic extension satisfying T. Of course, "large cardinal axiom" is
only slightly better than "natural theory": we have no general definition.
However, the consistency strength order on large cardinal axioms
corresponds to the inclusion order on their canonical minimal models. One
can abstract certain very basic properties of a canonical inner model
construction (roughly, one demands that it relativise to a real, and that
the model M(x) built over the real x have a wellorder uniformly definable
from x). One can then PROVE that any two such constructions are
comparable (in that for all reals x of sufficiently large Turing degree,
M(x) subset N(x) or N(x) subset M(x)). See "A classification of
jump operators", J.Steel, JSL, June 1982. I should point out here that
this result is proved from a determinacy hypothesis, so it is a case of
this way of climbing the consistency strength hierarchy proving something
in the direction of its own uniqueness. (The theorem is really just a
reformulation of Wadge's lemma, which can itself be used to make the same
point about inner model constructions in a slightly different way.)
John Steel
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