[FOM] On the Continuum Hypothesis, part 1

[Dmytro Taranovsky]

The Continuum Hypothesis (CH) asserts that every set of real numbers has
a bijection either with the set of all real numbers or with a set of
integers.  CH is the premier undecidable proposition in set theory.
Historically, it was the first problem independent of the axioms of set
theory (ZFC), and the first problem on Hilbert's celebrated list.  It
represents the problems that are independent through forcing, and thus
not decidable through large cardinal axioms as we know them.  Such
problems give rise to an argument that uncountable sets do not exist
because we do not have a plausible reasonable complete theory about
them.   

Of course, there are arguments for and against the Continuum Hypothesis.
However, to be accepted as a solution, an argument must either have
overwhelming supporting evidence, or provide a clear view of the theory
of arbitrary sets of real numbers.

In this posting, I will argue for the Continuum Hypothesis. In the
second part, I will present a particular argument, namely a general
picture of the power-set of R.  The totality of arguments may eventually
be considered a solution, but it is premature to do so now.

1. Some argue that the consequences of CH are unnatural.  However, the
intuition is based on canonical sets of reals, all of which are
measurable, have the property of Baire, etc.  It is a reaction to the
axiom of choice by trying to give as many nice properties to arbitrary
sets of real numbers as possible.
2.  The Continuum Hypothesis is in fact natural from the set-theoretical
point view, and the theory of "projective" sets of subsets of omega_1 is
more natural with CH.
3.  The theory of cardinal invariants has an effective version which is
meaningful under CH.  
4.  In its consequences, the Generalized Continuum Hypothesis (GCH) is a
natural strengthening of the axiom of choice.

5.  Some argue that CH is restrictive.  However, restrictive relative to
what?  We can maximize
(a) the totality of real numbers 
(b) the totality of subsets of omega_1 or sets of reals
Taken out of context, (a) seems to imply not CH.  However, since real
numbers are prior to sets of real numbers, we should maximize real
numbers first, and then maximize sets of real numbers (and subsets of
omega_1).  This favors CH.

6.  There appears no definable or otherwise natural set of real numbers
of cardinality strictly between omega and continuum.   In particular,
HOD(R) apparently satisfies the axiom of determinacy (AD), and AD
implies CH, at least as CH is defined above.  Negation of CH postulates
entire cardinalities of sets of real numbers, none of the sets
definable.
7.  All canonical fine-structural models--ranging from L to conjectured
models with subcompact cardinals and beyond--satisfy GCH.  In fact, a
fine-structural model is considered acceptable only if it satisfies GCH
(this is standard usage).
8.  Ground models for forcing usually satisfy GCH since GCH makes it
easier to work with sets and can be easily forced.
9.  GCH appears to be the only canonical theory of cardinal
exponentiation; no other (known) complete theory is remotely as
plausible.
10.  The Continuum Hypothesis is very easy to obtain through forcing.
In fact, whenever countably closed forcing adds subsets of omega_1, the
generic diamond is forced.  The generic diamond is a strengthening of
the diamond principle, and the diamond principle implies CH.  For
example, adding a generic subset of omega_1 (in this case, the poset is
simply a complete binary tree of height omega_1) always forces CH.
11.  Absoluteness under all forcing is too much to ask at this level of
expressiveness, but partial absoluteness favors CH.  Adding reals can
upset the structure of sets of reals, so a reasonable requirement is
that the forcing does not add reals.  Sigma-2-1 absoluteness under such
forcing implies CH, and CH + there is a proper class of measurable
Woodin cardinals implies Sigma-2-1 absoluteness under all forcing that
preserves CH.
12.  By 11, forcing axioms (such as the Open Coloring Axiom) can be
viewed as particular restrictions on which sets of reals are allowed.
Under CH (plus a proper class of measurable Woodin cardinals), any such
"axiom" that holds in a generic extension of V, also holds in an inner
model containing all the reals.
13.  Sigma-2-2 absoluteness under countably closed forcing implies the
generic diamond.  Sigma-2-2 absoluteness under all forcing for which the
generic extensions satisfy generic diamond is conjectured (by Hugh
Woodin) to hold assuming large cardinals.
14.  Woodin's Pmax Axiom implies absoluteness for H(omega_2) and not CH,
but is objectionable for several reasons:
(a)  It is not known to be consistent with large cardinals.
(b)  It implies that the theory of H(omega_2) has the same Turing degree
as the theory of H(omega_1) (second order arithmetic).  Thus, it appears
to minimize H(omega_2).
(c)  Under it, a well-ordering of the reals is definable from a subset
of omega_1, suggesting that |c| is really omega_1.

These arguments show that CH is probably true, but as it stands, are not
overwhelming.  We need a general theory of P(R).

Dmytro Taranovsky