# How large can the power set P(N) be made via forcing?

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.

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 $2^\omega=\kappa$.
• 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? – Thomas Benjamin Oct 13 '17 at 11:55