[FOM] Refuting CH?/6

[Harvey Friedman]

Continuing from http://www.cs.nyu.edu/pipermail/fom/2016-April/019774.html

I want to talk about the big picture.

CRITICAL ASYMMETRY BETWEEN CH AND NOT CH

I presented a very simple property of all Borel functions f:R into R
(provable in ATR_0). When lifted to all functions f:R into R, the
result is a statement provably equivalent to not CH over ZC.

QUESTION: Is there also a very simple property of all Borel functions
f:R into R sick that when lifted to all functions f:R into R, the
result is a statement provably equivalent to CH over ZC? Or even is
consistent with ZFC and implies CH over ZFC?

It appears that the answer to this Question is No. That the level of
complexity of such an example for CH will be orders of magnitude
beyond the level of complexity of my example for not CH.

I don't know of any way to establish this gross asymmetry between CH
and not CH except by building up the results along the lines of
http://www.cs.nyu.edu/pipermail/fom/2016-April/019774.html with richer
and richer languages.

A SIMPLE EXAMPLE WHICH BREAKS CHOICE

We give a simple property of all Borel functions f:r into R, provable
in ATR_0, which, when lifted to all functions f:R into R, results in a
statement refutable in ZC.

THEOREM 1. (ATR_0) Let f:R into R be Borel. There exists x in R and n
in Z such that |f(x) - f(x + e^n)| < 1.

PROPOSITION 2. Let f:R into R. There exists x in R and n in Z such
that |f(x) - f(x + e^n))| < 1.

THEOREM 3. Proposition 2 is refutable in ZC. It is provable in Z +
"every set of reals has the property of Baire".

AN EXPECTED PATTERN

It looks like if you start with very simple properties P of all Borel
f:R into R, then the following seems to be the case.

1. It is provable or refutable in ATR_0 that P holds of all Borel f:R into R.
2. When lifted to all f:R into R, the statement is provable in Z or
implies not CH over ZC.

In the above example, the lifted statement implies not CH over ZC
because it implies 1 = 0 over ZC.

This expected pattern is a gross asymmetry between CH and not CH. It
can be viewed as a kind of argument for not CH. I.e., not CH is being
chosen by "simplicity considerations".

Harvey Friedman