How kind of you to take an interest in my paper. Please see also
post about the dream solution and the arxiv entry for the paper.
First, I shall make a quibble, and then I'll address your question
at the end.
The quibble is that your quotation from the paper is not accurate.
The full paragraph from the paper reads:
I have argued, then, that there will be no dream solution of the continuum hypothesis.
Let me now go somewhat beyond this claim and issue a challenge to
those who propose to solve the continuum problem by some other
means. My challenge to anyone who proposes to give a particular,
definite answer to CH is that they must not only argue for their
preferred answer, mustering whatever philosophical or intuitive
support for their answer as they can, but also they must explain
away the illusion of our experience with the contrary hypothesis.
Only by doing so will they overcome the response I have described,
rejection of the argument from extensive experience of the
contrary. Before we will be able to accept CH as true, we must come
to know that our experience of the $\neg$CH worlds was somehow
flawed; we must come to see our experience in those lands as
illusory. It is insufficient to present a beautiful landscape, a
shining city on a hill, for we are widely traveled and know that it
is not the only one.
The difference is that it should say "extensive experience of the
contrary" rather than "extensive evidence of the contrary", a
difference that affects the meaning, since the point is that we
have experience in both the CH and in the $\neg$CH worlds. In
particular, there is a symmetry here, and I hope it was clear that
implicitly include your variation as part of my intended meaning.
Now, let me consider your final question, which is very good.
- Can one make the view showing that either Cohen reals are illusory,
or that the ability to add sufficient number of Cohen reals so as to make not-CH true is illusory, coherent?
I take the answer to be yes, these views are made coherent by what
I have called the universe view in my paper
The set-theoretic multiverse, from
which the dream solution paper is adapted. The universe view is the
view I am arguing against, and although I have attacked the
universe view for being mistaken, I do not attack it as incoherent.
The question is whether the alternative set-theoretic universes
that we seem to have discovered via forcing and other methods exist
as legitimate concepts of set or not. I have argued at length that
they do. But the opposing universe view is that no, there is just
one absolute background concept of set, and the purpose of set
theory is to discover what is true there. This seems to be a
perfectly coherent view. It is a view advanced explicitly by
Daniel Isaacson, who I quote extensively in my dream solution
paper, and also by Donald Martin, in his paper "Multiple universes
of sets and indeterminism in set theory", Topoi 20, 5--16, 2001,
Criticizing my argument, Peter Koellner has emphasized that one can
view my account of the naturalist account of forcing, rather than
providing evidence that forcing extensions are real, instead as the
desired explanation of the illusion of forcing extensions of $V$.
And perhaps this criticism is the detailed answer to your question.
That is, Koellner argues that the details of the proof of the
naturalist account of forcing is how one explains away the illusion
of forcing. So that would seem to be a coherent view. My reply to
that argument, in my multiverse paper, is that such an account of
forcing seems fundamentally crippling to our mathematical
intuition, if we must regard all talk of actual forcing extensions
of $V$ as ever-more-fantastical simulations of the extensions
inside $V$, something like the writings of the exotic-travelogue
writer who never actually ventures west of sixth avenue, or the
absurdity of the mathematician who insists that yes, the real
numbers exist with a full Platonic existence, but the complex
numbers do not; they must be simulated inside the reals, such as
with ordered pairs. The multiverse perspective makes a
philosophically simple position, taking the existence of the
forcing extensions at face value, while nurturing a robust use of
forcing that will ultimately aid our set-theoretical understanding.
Finally, let me say that I agree completely with Andrej's point
about geometry, and I discuss this analogy in section 4 of my