In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?
This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper:
"My challenge to anyone who proposes to give a particular, definite answer to CH is that they must not only argue for their particular 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 evidence of the contrary. Before we will be able to accept CH as true, we must come to know that our experience of the not-CH worlds was somehow flawed; we must come to see our experience in those lands as illusory."
Let me make a slight variation in the last sentence of his I quoted:
Before we will be able to accept not-CH as true, we must come to know that our experience of the CH worlds was somehow flawed; we must come to see our experience in those lands as illusory.
Since the goal of set theory (at least from Hamkins' perspective of the orthodox view (the set-theoretical universe as unique--it is the universe of all sets, "The set-Theoretical Multiverse: a model-theoretic philosophy of set theory")) is to have V (for ZFC, for example) to contain all possible sets short of inconsistency, it would seem that from this perspective that the CH worlds are already flawed and that to defend CH against not-CH one would have to say that the existence of 'Cohen reals' in the not-CH worlds is somehow illusory (or at least the belief that one can add sufficient number of Cohen reals to make CH false from the Naturalist View of Forcing perspective). 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?