[FOM] Refuting CH?/5

[Harvey Friedman]

Continuing from Refuting CH?/4

THEOREM 1. Let f:R into R be Borel.
i. There exist x,y such that x is no f(y+n) and y is no f(x+n).
ii. There exist x,y such that no x+n is any f(y+m), and no y+n is any f(x+m).
iii. There exist x_1,...,x_k such that no x_i + n is any fx_j + m).

THEOREM 2. Any of i-iii stated for all f:R into R is equivalent to the
negation of the continuum hypothesis, over ZC.

We view this as a special case of the Borel Transfer Principle:

ANY SIMPLE FUNDAMENTAL PROPERTY OF ALL BOREL FUNCTIONS LIFTS TO ALL FUNCTIONS.

At this early stage, we want to build some environments where the
Borel Transfer Principle is not only consistent with ZFC, but also
refutes the continuum hypothesis.

A moderately ambitious, but not very ambitious, environment, for this
early stage, is the following class of statements.

(for all Borel f_1,...,f_k:R into R)(there exists x_1,...,x_k in
R)(for all n_1,...,n_k in Z))(phi)

where phi is a conjunction of equations and inequations in terms using
multiple addition only with all but at most one summand being some
n_i.  We allow compounding of f's. I.e.,terms are defined inductively
as follows.

1. The variables shown are terms.
2. Any finite sum of 2 or more terms is a term, provided all but at
most one summand is some n_i.
3. The application of any f_i to any term is a term.

Note that every term has either no f's and no x's, or has no f's and
exactly one x, or has one or more f's and exactly one x. We call these
Z-terms, R-terms, and f-terms, respectively.

EXPECTED. There is an intelligible decision procedure for determining
the truth value of these statements. The statement "any such true
statement remains true with "Borel" removed" is equivalent to the
negation of the continuum hypothesis over ZC.

THEOREM. The Expected above is correct if we only allow inequations.
The decision procedure is that the inequations s not= t have the
following property. There is exactly one x_i, or (there are two
distinct x_i's and one or more f_i's, where all f_i's appear on one
side). Thus for each k, there is a strongest such Borel statement.
When lifted to all functions, it becomes equivalent to the negation of
the continuum hypothesis over ZC.

If we allow equations and inequations, we are going to get a lot of
interactions. However, I believe that these interactions are
manageably controllable.

This is all just a very basic early environment for Borel Transfer. My
feeling is that when Borel Transfer breaks - i.e., becomes
inconsistent - the environments are going to be comparatively
complicated. So SIMPLICITY is probably going to remain a driving force
in any satisfactory choice of new axioms according to this approach.

Harvey Friedman