Given the Constructable Universe L, is there a forcing extension L[G] of L in which P(N) is the 'size' (so to speak) of ORD, the proper class of all ordinals? I am interested in forcing extensions of L because L is the smallest submodel of ZFC containing 'all' the ordinals and as such, figuratively speaking, is the 'skeleton' of ZFC. The research program I am considering is using forcing to 'broaden' the power set P(A) of a set A so that P(A) contains all possible subsets of A (short of inconsistency), then using forcing to so broaden each stage of the Cumulative Hierarchy (insofar as the particular stage of the cumulative hierarchy can be so broadened) so that the 'true universe' V will contain all possible sets (short of inconsistency). I would be using the naturalist account of forcing. Is such a research program at all coherent? It would seem, at least, to be desired inasmuch as it seems intuitive to interpret the definition of P(A) ('the set of all subsets of A') as 'the set of all possible subsets of A'. Since no one[?] believes V=L, using forcing to broaden L seems quite natural.
1 Answer
With class forcing, yes, one can add ORD many Cohen reals, but the forcing extension will no longer satisfy the power set axiom and thus will not satisfy ZFC, although it will satisfy a significant fragment of ZFC. Indeed, this forcing is the one of the easiest ways to see that class forcing need not necessarily preserve ZFC. The most direct partial order to do this is Add($\omega$,ORD), the forcing to add ORD many Cohen reals, that is, the collection of finite binary partial functions from ORD to 2, ordered by extension. This is a proper class, and one must pay attention to the issues of class forcing, including the definability of the forcing relation. This forcing is the union of a chain of set-sized complete subposets, namely, the initial segments $\text{Add}(\omega,\theta)$ for cardinals $\theta$, and this is enough to let the basic forcing theory work out. But the power set axiom, of course, will not be preserved to the corresponding extension.
In general, since the ordinals of a forcing extension $V[G]$ are the same as the ordinals of $V$, if you are ever to pump up the power set of a set to become equinumerous with ORD, then that power set in the extension must be a proper class, and this is why you must lose ZFC in the extension.
Meanwhile, if you want to preserve ZFC, then you can use $\text{Add}(\omega,\kappa)$, and the basic fact is due to Solovay: for any cardinal $\kappa$ for which $\kappa^\omega=\kappa$ in $V$, and under GCH this includes any regular uncountable cardinal (and more), there is a forcing extension $V[G]$ in which $2^\omega=\kappa$.
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$\begingroup$ Thanks, Prof. Hamkins, that is helpful. I have another question that you in particular might be able to answer, since you co-wrote the the paper relating modal logic to forcing. The question is this: can the phrase 'all possible' be a coherent quantifier? As I mentioned in my question, I am interested in interpreting the universal quantifier in the power set axiom as 'all possible' subsets of a given set A. Of course the usual interpretation of the universal quantifier limits its interpretation to all sets in a given model(and such limitations give rise to the multiverse view) $\endgroup$ Mar 19, 2012 at 3:01
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$\begingroup$ But if one can coherently interpret the universal quantifier as 'all possible', even as a type of generalized quantifier, couldn't one have a set theory that satisfies the universalist demands, even though it would be different (slightly?) from ZFC (in the manner in which you specify)? $\endgroup$ Mar 19, 2012 at 3:11
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$\begingroup$ Question: Would the 'significant fragment' of $ZFC$ mentioned in this answer be, in fact, $ZFC^{-}$ + Collection? Also, could one replace "Cohen reals" with "random reals" in this result? $\endgroup$ Oct 13, 2017 at 11:55