[FOM] On the Continuum Hypothesis, part 2

[Dmytro Taranovsky]

In studying the Continuum Hypothesis, we reach a key question:

Is there a plausible reasonably complete theory of third order
arithmetic?  Is there a general principle that leads to such theory and
more?

Incompleteness in set theory is of two kinds:  Incompleteness through
forcing, and incompleteness from a lack of forcing-neutral set existence
principles.  The later incompleteness can be resolved through large
cardinal axioms, but sometimes the proofs are very difficult.  At
present, the impact of strong large cardinal axioms on P(R) is largely
unknown, and these axioms do not remove the incompleteness arising
through forcing.

However, there is a general maximizing principle that together with
reasonable large cardinal axioms gives (what we hope is) a reasonably
complete theory of sets constructible from sets of reals, L(P(R)).  I
should note that the principle is not yet generally accepted, and that
it is difficult to be certain about what is "maximizing" in the realm of
forcing.  

The principle is absoluteness under generic collapses.  For every
cardinal kappa, every possible subset of kappa exists, so if we try to
add more subsets in a reasonable way, the theory of subsets of kappa
(and somewhat beyond) will be unchanged.
	An example is Sigma-2 invariance of Theory(H(kappa^+)) under adding a
generic subset of kappa for every regular cardinal kappa.  It implies
GCH.

Adding a generic real without adding enough higher sets can disrupt the
theory of P(kappa):  For example, if we add a generic real and then a
generic subset of kappa, then kappa can no longer be weakly compact.
Instead, we assume that all sets in H(kappa) (for regular kappa) are
already included, and we maximize P(kappa) by collapsing higher
cardinals to kappa.  The three parameters in the principle are 
i.  which (infinite) regular cardinals kappa are allowed (perhaps all of
them).
ii.  which collapses are allowed (perhaps every Coll(kappa, <delta) and
Coll(kappa, delta) with kappa < delta).
iii.  the amount of absoluteness (absoluteness of the theory with or
without parameters in P(kappa) for perhaps L(P(kappa)) and as large
segment of the theory of P(P(kappa)) and beyond as is consistent).

The principle is open-ended and largely unexplored.  At the least, it
shows a coherent way to resolve incompleteness in set theory.  Some
statements following the principle are very strong. For example, (even
lightface) absoluteness of L(R) under every Coll(omega, kappa) and some
Coll(omega, <inaccessible) implies the axiom of determinacy in L(R). 

Question:  Assume that there is a (strongly) inaccessible cardinal. Is
it consistent for the theory of third order arithmetic to be invariant
under every Coll(omega, <inaccessible), Coll(omega_1, <inaccessible),
Coll(omega, kappa), and Coll(omega_1, kappa), where kappa can be any
uncountable cardinal?  If yes, is it consistent for the theory of finite
types to be invariant under all Coll(omega_n, <inaccessible) and
Coll(omega_n, kappa) (n ranges over natural numbers)?

To guarantee self-consistency, we settle on the following instance,
which eliminates the bulk of incompleteness about L(P(R)) in so far as
the incompleteness is achievable by forcing:

* There are (strongly) inaccessible cardinals alpha and beta with
alpha<beta such that (the real number coding) the theory of L(P(R)) is
invariant under Coll(omega, <alpha)*Coll(alpha, <beta).

(Here A*B means A followed by B; technically, it is a two-step
iteration.)  Assuming enough (to be precise, |c|^+ many) inaccessible
cardinals, this holds in a generic extension of V.  A slightly weaker
(but still sufficient) version can be forced from just two inaccessible
cardinals and does not imply existence of large cardinals: Statements
provable in ZFC\P about L(P(R)) intersect H(omega_3) after the collapses
are true.

The resolution implies the diamond principle for every stationary subset
of omega_1.  (This principle also holds in L, which explains a certain
similarity with the theory of L.  Because L is canonical, certain
independent true principles hold in L.)  It also negates the Kurepa
Hypothesis in a strong way:  Every set in L(P(R)) of subsets of omega_1
either has cardinality at most omega_1 or contains a perfect subset
(these are defined using perfect trees).

The first collapse is included primarily to get the most without large
cardinals.  Just Coll(omega_1, <beta) should suffice under large
cardinal assumptions.
	Conjecture:  Under large cardinal assumptions, the theory of L(P(R)) as
computed in Coll(omega_1, <delta) where delta is inaccessible is
reasonably complete.

What large cardinal strength is sufficient is not yet known, but a
reasonable guess would be existence of a supercompact cardinal.  A pure
large cardinal axiom should be sufficient if it implies that in
Coll(omega_1, <delta), no well-ordering of the reals is ordinal
definable in L(P(R)).  At the Sigma-2-1 (and probably Sigma-2-2) level,
generic absoluteness is enforced by universally Baire sets of reals,
which means that the absoluteness holds with real parameters from every
generic extension of V.  At substantially higher levels, generic
absoluteness is enforced by sets of sets of reals (and beyond), but we
do not know how that works (or how to prove such absoluteness).  We
conjecture absoluteness of Theory(L(P(R))) (without parameters) as
computed in Coll(omega_1, <inaccessible).  In any case, we can study the
theory of L(P(R)) under Coll(omega, <alpha)*Coll(alpha, <beta) without
waiting for these results.

An illustrating example is the theory of L(R) in ZFC under Coll(omega,
<inaccessible), and how the theory computes much of the truth about
L(R).  An inaccessible is used because we have to maximize R relative to
every real number included, not just those in the ground model.  As is
clear from the proofs, the theory obtained is nice because of symmetry.
The actual mathematical universe is maximal and complete, so there is no
room for asymmetry.  That symmetry by itself suffices to prove that
every projective set is measurable.  Not only does the boolean algebra
for Coll(omega, <inaccessible) have a high degree of symmetry, but it
also erases asymmetries produced by pathological forcing:  If |
P|<inaccessible, then P*Coll(omega, <inaccessible) is equivalent to
Coll(omega, <inaccessible).  Analogously with countably closed forcing
and Coll(omega_1, <inaccessible).

Through forcing, gross asymmetries in P(R) can be produced.  For
example, for every set of reals S, there is a generic extension of V
which does not add reals and in which S is Sigma-2-2 definable without
parameters.  Symmetry through maximality is the key reason for the power
of generic collapses.

Just to note, the closest a generic extension of L comes to being true
about L(P(R)) is through Coll(omega, <alpha)*Coll(alpha, <beta), where
alpha<beta are Silver indiscernibles.

We conclude with a sociological perspective:  What behavior is being
advocated?  While the study of various generic extensions is important,
the theory proposed, apparently being true, holds a privileged position
among them.  The theory should be studied, and whenever an important
incompleteness in L(P(R)) arises through forcing, a question to ask is
whether it holds in the theory.

There is a continuum of statements ranging from interesting hypotheses
to those we apply without even noting their use.  The solution presented
to the Continuum Hypothesis (if it is a solution) can be considered as a
semi-axiom as it is our best but not certain guess about which
statements are true.  For formalists, I note that truth seems to
coincide with naturalness and beauty, something that even formalists
admire.

Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm